I am nowhere near fluent enough in math to calculate this problem. I went with 49^51 being larger based on one simple thought.... If you multiply 50 by itself 50 times you get a number that is astronomically huge. If you then multiply 49 with itself 50 times you get a smaller number that is also astronomically huge. But that smaller number will then get multiplied with 49 one more time, which will be a much bigger number. Then, to test my idea I tried smalle numbers. 10^10 is 10 billion. 9^11 is more than 3 times higher.
@normalone91996 місяців тому
Same here 🎊
@alessiosandro1235 місяців тому
The thing is i guess that with higher potencies it gets bigger
@HashiRa2485 місяців тому
Sometimes the journey is more important than destination.
@zdenekbina60445 місяців тому
I thought the opposite. 50 is larger base, so you can calaculte the 1.02 to the power of 50. If its larger than 49, its bigger. If not 49^51 is bigger.
@zdenekbina60445 місяців тому
And that is 2.69 so it supports your output. You got it right. My methid confirmed your result.
@rchatte1002 місяці тому
Logically, the power is always the most powerful part of a number.
@OblomSaratov2 місяці тому
It usually is, but it's not always the case. For example, 4^4 > 3^5.
@GHOST-RIDER-0Місяць тому
2¹ > 1∞
@OblomSaratovМісяць тому
@@GHOST-RIDER-0 that's not true because infinity is not a number and 1∞ is undefined.
@hafidmostarhfir2245Місяць тому
Only if u are powering numbers greater than 1 ..I think
Місяць тому
@@OblomSaratovthen 1^999999
@chesfern4 місяці тому
The simplest solution would be to let 50=n, 49=n-1, and 51=n+1. Taking logarithms of both numbers: Log n^n = nlogn and log (n-1)^(n+1)=(n+1)log(n-1). For a large number n, log n is approx. =log (n-1). Therefore, we are left with 2 values n and n+1 and obviously. n+1 is greater than n, which is why 49^51 is bigger than 50^50.
@jackwilson55422 місяці тому
The easiest solution is comparing 4^5 vs 5^4. There is no reason why the trend won't continue with higher numbers.
@wolf53702 місяці тому
@lson5542 That doesn't match the pattern in the Q. 5^5 vs 4^6 would.
@konglink3359Місяць тому
@@jackwilson5542 but then u have to prove that it's true for all numbers, with concrete evidence
@tontonbeber4555Місяць тому
@@konglink3359 only if numbers are > e ... that's the only condition. It doesn't work for 2 and 3 for instance ...
@grapefruitsyrup8185Місяць тому
@@tontonbeber4555 but why e? That's so specific how do you figure the explanation?
@mingwangzhong1176 місяців тому
We need to compare f = (n+1)^(n+1)/n^(n+2) with 1 for a large n. This expression can be rewritten as f = (1+1/n)^n * (1+1/n)/n. Then using the standard binomial expansion, we have (1+1/n)^n = 1 + {n}*1/n + {n*(n+1)/2}*1/n^2 + {n*(n+1)/3!}*1/n^3 + ... < 1 + 1 + 1/2! + 1/3! + 1/4! + ... < 1 + 1 + 1/2 + 1/4 + 1/8 = 3. Therefore, for a large n such as 49, f should be smaller than 1, i.e., 50^50 < 49^51.
@emreyukselci5 місяців тому
A simpler solution: 49^51 / 50^50 = 49 x (49/50)^50 If take the square root which needs to be over 1 if 49^51 is larger we get: 7 x (49/50)^25 . And (49/50)^25 > 49/50 - (25x1/50) as each power of 49/50 will reduce the number less than 1/50. So (49/50)^25 > 24/50. Then, 7 x (49/50)^25 > 7 x 24/50 > 1. As a result 49^51 > 50^50 .
@user-qo5jo2qc5q2 місяці тому
Now this is a great solution, the video's solution is less elegant as it depends on students having memorized the definition of e. This solution however only relies in algebraic manipulation. It was a bit hard to follow, I originally thought you were wrong there at the end, but upon further analysis, indeed you have proven it! Well done, thank you for sharing this solution, much better than the video's.
@young47832 місяці тому
Good one!
@dragondompyd71712 місяці тому
Maybe I don't understand it fully but as you said, we can only use the squareroot while holding inequality if the value (49^51/50^50) is greater than one. And since that is exactly the thing we are trying to prove, we can't assume that it is greater than 1.
@santoshkumarvlogs37532 місяці тому
Nice solution
@alexbayan8302Місяць тому
This is a very nice "high school" solution. That is to say you will get a high mark if you wrote this solution in a Math Olympiads. But conceptually there are simpler solutions;just binomial expansion.
@aakashanantharaman40375 місяців тому
Thanks for the fun mental challenge, and sharing the lateral thinking and inference thinking that goes with it. I enjoyed watching your video ;)
@user-hz5ne2rl5e5 місяців тому
You need to set up an axillary function to analyse the rate of the fuction at a point. 50^50 ? 49^51 ln(49)/(49+1) ? ln(50)/(50+1). Then axillary f(x)=ln(x)/(x+1). Find the derivative and approximate extremum point. Maximum of f(x) is at x on (3, 4). f(x) at x=49 is monotonic and decreasing. Hence, ln(49)/50 > ln(50)/51 49^51 > 50^50 .
@vladpetre56746 місяців тому
You have to analyze the function lnx/(x+1). This has a max value when x is somewhere between 3 and 4 (where its derivative is equal to 0), after that decreasing but staying > 0. So for any x>4 we have lnx/(x+1) > ln(x+1)/(x+2). Making x = 49 we have (ln49)/50) > (ln50)/51, thus 49^51 > 50^50.
@epevaldon54215 місяців тому
Oh i got head ache on math. Im so poor on math
@romain1mp5 місяців тому
Thanks for that! The demonstration on the video is not acceptable (unless if the goal is only to get the most probable answer without caring about how you get there). Using a limit to justify an inequality is not sufficient at all! Your method (even though I didn’t verify it) is more rigourous..
@theupson5 місяців тому
@@romain1mp the vid is easily rehabilitated; (1+1/49)^49 is immediately less than e because discrete compound < continuous compound (for positive r)
@lizekamtombe22235 місяців тому
@romain1mp That would make the squeeze theorem wrong and also disprove Archimedes, strict inequalities are very useful proofs, but they have to be strict. "Equalities are for children, real men deal with inequalities!"
@romain1mp5 місяців тому
@@lizekamtombe2223who talked about equalities here? I am just saying that if Lim(f(n)) is smaller than L … when n is close to infinity…. You cannot conclude that f(n=49) is smaller than L without more inputs… For example you need to demonstrate quickly that f is increasing function from a certain level p (i.e. p
Take the logarithm of both numbers. For numbers above zero, a > b if log a >log b. Taking log of both sides reduces the problem to 50 * log 50 which is between 84 and 85 51 * log 49 is between 86 and 87
@michaelhartmann12855 місяців тому
That was the first approach that crossed my mind, and a great deal simpler than the algebraic thicket the narrator lays out.
@kanwaljitsingh32485 місяців тому
Good solution
@hrvat77705 місяців тому
But how do you calculate log50 and log49 without a calculator, to come to the conclusion the right side us larger then the left one? I'm sure the point is to solve the problem without a calculator as otherwise you can just calculate both initial terms and see which number is larger 😉
@justanotherguy4695 місяців тому
I'm not that well versed in mathematics. Is it a proof, though?@@michaelhartmann1285
@srinathparimi335 місяців тому
By using properties of log, we can write log50 as log 5 + log 10 and log49 as 2log7. Now, log5 and log7 values can be approximately substituted. log5~0.698 and log7~0.845
@Arunmsharma5 місяців тому
Use compound interest logic, you’ll do this much earlier. 50/49 is 1.02 approx raised to 50, will be much below 49, think of it as getting a 2% interest on your bank deposit. It will take maybe 25-30 years to double, and would max be 4 times in 50 years - hence 50/49 raised to 50 is def below 4, and when you divide by 1/49, it’s clearly below 1.
@user-qo5jo2qc5q2 місяці тому
That's not a mathematical proof, you can have a gut feel (if you have worked a lot with interest) that your answer is right, but that's not what this question is about.
@ckshene72126 місяців тому
No, you still have to prove (1+1/49)^49 < 3. This is because the limit may or may not be monotonically increasing or decreasing. Therefore, you have to prove the limit is monotonically increasing to justify your claim.
@thanhquenguyen94626 місяців тому
Agree. Without proving that we can doing the same way as the video to approve 4^4 < 3^5, but it not.
@supergaga16 годин тому
My intuitive solution is: Let x = 50, and apply natural logarithm on both sides, we have x*ln(x), and (x+1)*ln(x-1). Now look at two terms: (x+1) / x and lnx / ln(x-1), they are just gradient of functions y=x, and y=ln(x) respectively, with dx = 1. Obviously y = x has constant gradient of 1, while y=ln(x) has decreasing gradient (always < 1) approaching to zero. Thus we have (x+1) / x > ln(x) / ln(x-1), therefore, x*ln(x) < (x+1) / ln(x-1). so we have proved 50^50 < 49^51 in a very simple way.
@ducngoctd6 місяців тому
Chứng minh được mệnh đề tổng quát, (bằng phương pháp quy nạp toán học): n^n > (n+1)^(n-1). Với n = 50 là bài toán mà bạn nêu ở trên..
@Ben-pw3qe5 місяців тому
By using Log you come to the answer in a few Seconds; Log 50^50 = 50 Log 50 = almost 85 Log 49^51 = 2 x 51 x Log 7 = almost 86 So 49^51 is almost 10 times bigger than 50^50 😊
@neiljohnson79145 місяців тому
But you need to use a calculator. This problem asks that you come to a solution according to math principles, not raw calculations
@user-bo5fi7os2p5 місяців тому
I knew this without even calculate anything lmao
@chaplainmattsanders48842 місяці тому
i don’t understand that, but I believe you!
@vandemaataram26002 місяці тому
Yes. Your solution is similar to mine.
@TheSimCaptain6 місяців тому
Its much simpler than the video. You find the 50th root of both numbers. That means you divide each exponent by 50 such that you get 50/50 and 51/50. Then you just calculate 50 to the power of 50/50. And then 49 to the power of 51/50. 50 to the power of50/50 is 50 to the power of 1 or just 50. 49 to the power of 51/50 is 49 to the power of 1.02 which is 52.966. Because the fiftieth root of 49 to the power of 51/50 is bigger, then that number must be bigger.
@manny20926 місяців тому
I like this answer already!
@kaustubhprakash12736 місяців тому
This is a good answer. However, you would need to show how 49^(1.02) = 52.966. For this you can do it by binomial expansion and it should be easy to do
@mareshetseleshi27173 місяці тому
Much appreciated
@donmoore77852 місяці тому
How did you calculate 52.966 - a calculator?
@TheSimCaptain2 місяці тому
@@donmoore7785 Yep.
@Chawlas573 місяці тому
Your voice is very sweet to listen... Loving and enjoying your voice
@tassiedevil22007 місяців тому
Since you appeal to recognising Euler's number as the limit as n-> infinity, you need to show that it approaches that limit from below to use it as a bound for finite n=49 case.
@phajgo27 місяців тому
which is actually quite obvious..
@user-gk3on7xp7e7 місяців тому
@@phajgo2 No, it isn't obvious! Unless you have bounds of the accuracy. On the other hand, the result is correct because it is well known that the sequence (1+1/n)^n is strictly increasing.
@phajgo27 місяців тому
@@user-gk3on7xp7e exactly this is why I think it is obvious. For n=1, (1+1/n)^n = 2, for n=2, (1+1/n)^n=(3/2)^2=9/4=2,25. and at infinity we know it is e so it's approaching from the left. Isn't that enough?
@tassiedevil22006 місяців тому
@@phajgo2 - I guess that provided one adds @user-gk3on7xp7e insight and says "the limit as n-> Infinity is Euler's number and it is well known that the sequence is monotonically increasing" you'd deserve the marks? This rather makes it a general knowledge test in my opinion. One can use calculus to prove that (1+1/x)^x is monotonically increasing for x>0. I'm not convinced that listing the first few terms of a sequence is a proof, although you seem actually in furious agreement with @user-gk3on7xp7e and the "it's well-known" proof by assertion. I looked at raios of successive terms and didn't see a quick proof of monotonicity. Just my 2c.
@mikaelhakobyan93636 місяців тому
@@phajgo2 No, it isn't. A function can increase at first, then start to decrease, and after that move to it's limit.
@user-hb1vf6lo7p2 місяці тому
Spasibo. If you can't wait, quickly in python: print(len(str(50**50)), len(str(49**51)))
@user-ec8ru7je7bМісяць тому
why len? int же
@user-hb1vf6lo7pМісяць тому
@@user-ec8ru7je7b привет. len посчитает количество символов в каждом из представленных(чтобы не выводить большие числа). Какой из вариантов больше, будет более очевидно.
@madankundu6035Місяць тому
that's cheating... ha ha
@tonybantu16815 місяців тому
Should have applied natural logarithm rule. Very easy... ln( 50^50 ) ...?... ln( 49^51 ). Take any log of both sides. 50x (ln 50 ) ...?... 51x ( ln 49 ) ---> 50/51 ...?... ln 49 / ln 50 ---> 0.98... < 0.99... (Reduced to numeric 2-decimal places on both sides). Or 50/51 < ln 49 / ln 51. Therefore: 50^50 < 49^51 No need for advanced formulas or calculus. Just logarithm rule and basic arithmetic. Someone was close to this but complicated the simplicity of basic power rule of logarithms with functions, extrema, etc. All exponential power comparisons are pre-calculus algebra or arithmetic. Variations, infinitesimals or their limits are unnecessary.
@jarl34345 місяців тому
I ended up here on an insomniac night (very counterintuitive) and this is just the perfect voice I needed to calm down and have a good night. Thx!
@billj56456 місяців тому
maybe simplify this greatly- which is smaller- 3 to the power of 3, or 2 to the power of 4? (or 4 to the power of 2). 3 to the power of 3 is the smallest so must be a minimum point if you graphed these
@akhan99692 місяці тому
Good one, thank you for sharing.
@MathsMadeSimple1017 місяців тому
Great explanation
@TheSoteriologist5 місяців тому
Finding an unnecessarily complicated, inelegant and difficult solution is not a sign that one should be a mathematician.
@Aeyo3 місяці тому
Your voice was soothing and gave me peace while my mind was screaming inside
@zeroun925 місяців тому
I took the natural log of both sides and saw that it was 50 ln 50 vs 2450 ln 49. This showed a clear difference. The other method suggested is a more general approach that I didn't think of. Works better in the long term.
@texasaggiegigsem5 місяців тому
I did the same thing, but I like her approach without having to find the natural log, which I reached for my calculator for.
@AbhishekChoudharyB4 місяці тому
Shouldn't it be 51 ln 49 How did u get 2450?
@GoodChemistry5 місяців тому
I love these problems, great mental exercise! Thanks.
@TheRootOfJoy5 місяців тому
Beautiful explanation! Thanks
@karthik999x-narrowone83 дні тому
I don't care about what your teaching. I just came here cuz your voice is soothing and your handwriting is crystal clear.
@gibbogle6 місяців тому
You didn't prove that (1 + 1/m)^m < lim (1 + 1/n)^n as n -> infinity. It's true, but you assumed it without proof. You need to show that (1 + 1/n)^n < (1 + 1/(n+1))^(n+1) for all integers n > 0.
@tharock2206 місяців тому
That's a good point. Maybe take the derivative and show it's always positive? Either way it's a good explanation and doesn't require a calculator.
@thomasdalton15086 місяців тому
@@tharock220The derivative is rather messy, so I don't think it will be easy to prove it is greater than zero, but I expect it is possible.
@bumbarabun6 місяців тому
@@thomasdalton1508 I believe that derivative is n * ( 1 + 1/n ) ^ (n-1) and it cannot be negative for positive n
@thomasdalton15086 місяців тому
@@bumbarabun Why do you believe that? You are differentiating with respect to n, so you can't use the rule for differentiating x^n with respect to x. The variable we are differentiating with respect to appears in both the base and the exponent, so it is a complicated differentiation. It's like differentiating x^x, but worse.
@bumbarabun6 місяців тому
@@thomasdalton1508 you are right, my mistake
@iviewthetube6 місяців тому
Thank you for explaining such a terrifying problem so calmly.
@JH-pe3ro2 місяці тому
Since I've lost access to my logarithm skills I took the approach of graphing the first terms in the series to observe convergence. At first it's unclear with 1^3=1 vs 2^2=2, but 2^4=16 narrows with 3^3=27, and again 3^5=81*3=243 vs 4^4=16*16=256. Intuitively, as n grows, the relative effect of the larger exponent will overtake the base value - 2^64 is much more than 64^2. Since the problem doesn't call for a more specific answer, it can end after plotting an estimate of the crossover. I plugged it in a graphing caculator to check and it's very obvious that before n=5 you've already crossed over.
@zahariastoianovici8590Місяць тому
This is an excellent problem and great way to resolve , did learn a lot
@crannogman62896 місяців тому
Write 49^51 as (50-1)^51 and write out the first few (5 should do it) terms of the expansion. Use combinatorics to find the coefficients. Rewriting it in terms of 50^50 gets you to see its about 18*50^50.
@larswilms82756 місяців тому
I rewrote 50 as 49+1, since this gives only positive terms. But same reasoning. end up with it being less than 2*49^50 so the fraction would be less than 2/49 which is less than 1 so 49^51 is greater
@chris85356 місяців тому
Yea this is super obvious with just logic. 50^50 is less than (50-1)^51. Like 2 seconds
@MauuuAlpha5 місяців тому
I did something similar
@GarryBoyer2 місяці тому
The binomial expansion would be a good solution if it were more of a close call. But if you have done enough problems like this you can quickly recognize the limit to e and come up with an answer more quickly without doing much math.
@TheThrakatuluk6 місяців тому
6:53 How did one < times one > times one < ended up as one < symbol? Does it only bother me?
@oo_rahbel_oo5 місяців тому
Well done. It's pretty clever how you've managed to prove it
@vandemaataram26002 місяці тому
Take 'log', then the problem wil become too easy. 👍👍👍 But I think, the problem is of arithmatic. That's why we are having these complicated solutions.
@skhadka2466Місяць тому
Its so simple left side become (49+1)^50 then on simplification it becomes (49^50)+1^50 direct compare to righ side which is greater than left side.
@atulyaroy89626 місяців тому
You can see by binomial approximation that 49^51 is greater. Though I believe the gap is big enough that the error wouldn't matter. Since LHS will have factor of around 2 and RHS will have factor of 49 which is quite large.
@sorinturle45995 місяців тому
When the numbers are close, the exponent (power) beats the base (the number). In fact, the bigger these numbers are, the smaller number with the bigger exponent (only by one unit) can go lower and lower from 50% of the higher number and the exponentiasion will be higher. Of course, this is only the answer, not the demonstration. More rigurous, but still simple is using the logarithms.
@atulyaroy89625 місяців тому
@@sorinturle4599 well even at near x=0 there is crazy separation for exponential graph with higher bases so it makes sense differing by 1 in base doesn't matter as much differing by power by 1
@Skaahn6 місяців тому
My simple approach to guessing, just simply the problem as (50)^1 = 50 and (49)^2 = 2401, so RHS will be bigger
@rahuldwivedi47586 місяців тому
If you needed to apply Euler’s, why did you need to extract (1+ 1/49)^49* (1+1/49) Wouldn’t that hold true directly for (1+1/49)^50?
@rangarajanvenkatraman7627 місяців тому
Nice solution😊
@Tomaslyftning6 місяців тому
Divide both sides with 49^50. 50/49 is about 1.02 therefore we can compare 1.02 ^50 with 49. Rule of thumb: to double 1.02 we have to multiply it 35 times so 1.02^50 is less than 4. Since 4 is less than 49 we get the same answer as the video.
@herotb2210916 місяців тому
Wow. I love it
@Aut0KAD6 місяців тому
nice, using the rule of 72?
@mujtaba216 місяців тому
That's how I thought of it. You wrote it concisely 👏🏽
@mujtaba216 місяців тому
@@Aut0KADyes
@Btitude6 місяців тому
I found my think-a-like buddy. I thought exactly the same way applying the rule of 72. I thought no one else would be smart enough. You are really smart, mate.
@GetMeThere12 місяці тому
Thanks for doing this for "n." So (for future reference) we know that n^n < (n-1)^(n+1) for relatively large n. What is the lower cutoff (using integers), where the inequality sign switches? 4^4>3^5 but 5^5 < 4^6
@MonsterERB6 місяців тому
Move everything to one side and you're asking "Which is bigger: (50^50)/(49^51), or 1?" Then that simplifies to [(50/49)^50] versus 1... which is "number very close to e, divided by 49" versus 1. Pretty obvious then than 49^51 is larger.
@robertoguerra53755 місяців тому
The ratio of the 2 numbers would be approaching n/2.72 Less than 1 for n>2
@rasikparray5575Місяць тому
I think there's no need for such a laborious approach... We can get this done in two steps Step one divide both terms and multiply and divide by 50 Step two (50÷49)^51 /50 is final expression which is clearly< 1
@sketchwarehelp4 місяці тому
Dividing 50^50/49^51, if numerator is larger then it will be >1, otherwise it will be
@amitgp20074 місяці тому
Your pen is awesome. Which one brand?
@user-vo2zo9jh3o7 місяців тому
I wrote the equation as: 50^50 / ( 50-1)^50 = 50-1, then, replace 50 by X. But still cant solve, it any advice?
@RikMaxSpeed6 місяців тому
That looked like a very long complicated approach: I took a log on both sides and approximated ln(50) = ln(49)+1/49 (ie: first order derivative and taylor series). Way simpler!
@thomasdalton15086 місяців тому
If you are being rigorous, you would need to put bounds on the error in the Taylor approximation and show that they can't change the conclusion. You can certainly do that, but it gets a little messy. (Doing it the way in the video, you need to prove that (1+1/n)^n is monotonically increasing, which also isn't straightforward.)
@EdwardCurrent5 днів тому
I just started with small numbers and gradually incremented them. By the time you get to 5^5 vs. 4^6, the 2nd evaluation is outpacing the first, so you can safely assume the trend will continue for numbers up to 50.
@stem2-orgayafelizardoiiiy.9Місяць тому
By Modular Arithmetic for divisibility of 50 50^50 ___ 49^51 0 ____ (-1)^51 0 ___ (-1) 0 ___ 49 (since the remainder must be positive) 0 < 49 therefore 50^50 < 49^51
@antoniojunior9365 місяців тому
Eu assisti em outro idioma e entendi, por isso eu amo a matemática ❤
@arthurhairumian7179Місяць тому
I solved the problem in 2 seconds with my intuition ...and the answer was correct, so it's true that imagination is more important than the knowledge - Einstein
@d8ngdeld8ng3 місяці тому
Same answer result. But it glaringly shows how Mathematician and an Economics and Finance pips answered this numerical logic query step by step relative to their learned principles.
@Greasyhair4 місяці тому
You generalize (x-1)^(x+1) / x^x and take limit at 1 and infinity. You took see it's a diverging function and hence 49^51 > 50^50. Or in general any (x-1)^(x+1) > x^x for x >>1.
@mda77636 місяців тому
And if there’s anyone who knows a harder way to do this, the ball is in your court now
@RikMaxSpeed6 місяців тому
Spot on, this was way over-complicated.
@88kgs6 місяців тому
😂😂
@daou63Місяць тому
good explanation proffessor
@axeldep.14585 місяців тому
Just compare logarithmic values. 51 ln 49 vs 50 ln 50 51 ln 49 vs 50 (ln 49 + ln 50/49) ln 49 vs 50 ln (50/49) ln 49 vs 50 ln (1+1/49) Right hand side is smaller than it's first degree approximation since ln (1+x) is concave. So right hand side is smaller than 50/49, which is way smaller than ln 49 = 2 ln 7 > 2 since 7 > e. Left hand side is way bigger. So 49^51 is bigger than 50^50.
@Palisade58102 місяці тому
Since 1/49 will diminish rapidly in the binomial expansion of (1+1/49)^50 you can approximate it to 1+50/49= 2+1/49 to the first order so (50^50)/(49^51) will be (2*49+1)/(49^2)=99/2399
@lagautmd5 місяців тому
This is much simpler to analyze. Make it easily envisioned by reducing the bewilderingly large powers. Get them down to human scale. 50^1 compared to 49^2. 50^1=50. 49^2 is bigger than 50 by inspection. Therefore, by induction, n^n < (n-1)^(n+1).
@aniruddhshandilyak32895 місяців тому
A good analysis But this works only when n>=5
@user-zo1bv7hk1l5 місяців тому
right hand side is increasing at a slower rate than the left hand side, so at some point when n increases, left hand side should overtake the right hand side
@notray24456 місяців тому
It’s easy, man. For all x,y,a,b > 1: x^a > y^b if a>b AT ALMOST CASE
@joshuavasquez90192 місяці тому
freaking mic drop at the end there. sheesh. Super cool video, thanks for making it!
@jotabi5526 місяців тому
Thanks for the video. Regards
@jakobullmann75864 місяці тому
The log derivative of (a-x)^(a+x) wrt x is -(a+x)/(a-x)+log(a-x). This is greater than -51/49+log(49) for a=50 and x=0,…,1, which is much larger than 0 as can be easily estimated (log(49)>log_3(49)>log_3(27)=3, for example). Hence monotonously increasing, hence 49^51 is greater.
@shoodler3 місяці тому
I tried making a graph of this relation in the form of x^y_(x-1)^(y+1). For values of x from 1 to 4 (integer) LHS will always be greater (regardless of value of y) and for any nunber greater than 4 RHS will always be greater regardless of the value of y
@shoodler3 місяці тому
The underscore represents a blank for what their relation is
@JagatK_5 місяців тому
I knew the answer but i couldn’t stop the video in between because of that voice. That damn voice ❤
@korathmathew5 місяців тому
beautifully explained.
@jackmclane18265 місяців тому
Higher power wins for all numbers > 5^5 49^51 is actually almost 20 times larger than 50^50. (bruteforced it by excel)
@OblomSaratov2 місяці тому
Higher power doesn't always win. 4^4 > 3^5.
@pengzhang-sy6zwМісяць тому
wa oh! your voice is so good, It feels like I am hearing asmr. And it helps me to sleep.
@GunjanSharma-nf2ce3 місяці тому
I went with binomial expansion by splitting 49^51 into (50-1)^51. Then in its expansion we have 50^51 with the additions of many more digits.
@kgfes6 годин тому
To compare which number is larger between 50^50 and 49^51, we can calculate their values. 50^50 ≈ 8.881784 × 10^69 49^51 ≈ 1.319507 × 10^77 Comparing the two values, 49^51 is larger than 50^50.
@javanautski3 місяці тому
Interesting. If you use 4^4, that's > 3^5, but starting with n=5, we have n^n < (n-1)^(n+1). I think the left hand is Lambert's w-function?
@JohnsOnStrings2 місяці тому
Cool trick. Important to remember defn of e.
@davidsousaRJ2 місяці тому
I expanded 50^50 as (49+1)^50 using the binomial theorem and compared each term with 49^51 which is 49^50 + 49^50 + ... + 49^50 (49 times). It ends up like each term in the second expression is larger than the ones in the first expression.
@GMBeaulac6 місяців тому
I’m not sure why you back away from the numbers at the end. Sure it’s less than 1, intuitively one might have guessed that a small difference in powers will have more of an impact than a small difference in bases. But without that simplification you’d have an idea of how large the difference actually is. 3x50 (150) over 49 squared isn’t that bad; square 50 (2500) then subtract 50 to get 50x49 then subtract 49 to get 49 squared, or 2500-99 or 2401. It’s still not exact by any means, since the 3 is an approximation, but you can easily tell at a glance then that 49^51 is between 15 and 20x larger than 50^50
@thenamedoesnotmatter2 місяці тому
I used simple intuition. It makes more sense to me that the (49x49 ... x49) falls behind (50x50... x50), however the extra instance of multiplying x49 accounts for all of the previous distance between those equations. If you are repeating something 50x, and 1x49 is just 1 less than 50. We haven't gotten far enough exponentially to create more than 1 digit of a gap. We know from multiplication rules it will go around 2x10^20 for either equation, but it just intuitively makes sense that 49^51 > 50^50.
@vivekmittal23Місяць тому
49 could be expressed as 50 ,-1 and then apply binomial expansion. Take the ratio of given two terms and can be solved easily. Just see whethr ratio is less than or greater than 1
@rogerahier47506 місяців тому
This is true for any x^y and (x-1)^ (y+1). Well, for positive numbers greater than 1. Didn't look at others, but probably reversed for others.
@adamfarmer76656 місяців тому
when x>5 that is, not when x>1
@robertoguerra53755 місяців тому
@@adamfarmer7665that is a good math question: prove that ALL x^x < (x-1)^(x+1) for x > 5
@DataHash5 місяців тому
I think it's simpler using binomial expansion of 49^51 = (50 -1)^51 = 50^51 + other stuff and therefore > 50^50
@N7_YES2 місяці тому
Awesome mam
@mamcsoft20122 місяці тому
I did a really simple equation, whats bigger 50x50 or 49x49x49, I deducted that it would deviate acordingly, and apparently it works for any positive number greater than 1
@naseeb.shalimar4 місяці тому
Take (a - b) and divide both sides by 50^50. That is (50^50)- ((50-1)^(50+1)) = x. Divide x by 50^50.. You will get x as negative meaning the latter is bigger
@joker78785 місяців тому
How can we confirm that (1+1/n)^n is monotone increasing function?
@curaticac53916 місяців тому
This is flawed like many such math "solutions" on UKposts. The fact that the limit of a sequence is < 3 does not guarantee that the terms of the sequence are equally so. Like other viewers pointed out.
@nomusic11796 місяців тому
basically doing log at the end. you could have done the log in the beginning and solve it early. 50*log(50) vs 51*log(49)
@TravelingMooseMediaДень тому
Pretty easy. It’s easy to calculate in your head that 4^6 > 5^5. 4096 > 3125. So for any number greater or equal to n = 5, n-1^ n+1 > n^n.
@eekeek46382 місяці тому
(5×10)^50 on one side and (10×4,9)^51 on other side. Now we silplify by 10^50 and we get 5^50 and 10x4,9^51. Next 4,9/5 is 0.98. Then we have 10x0.98^51x5x5^50. 50x0.98^51 is greater than 1 so 49^51 is more than 50^50.
@gulboyrathesungod5 місяців тому
49^51. There is one additional zero there. Triumphs the other easily. No need to complicate the problem.
@peterectasy29575 місяців тому
could that be extended to all numbers ? this example : 50 over 50 versus 49 over 51 general example : n over n versus (n-1) over (n+1)
@OblomSaratov2 місяці тому
As a calculus fan, to solve this I analyzed the function y=(50-x)^(50+x), where y(0)=50^50, y(1)=49^51. Its derivative is y'=(50-x)^(50+x) (ln(50-x)-(50+x)/(50-x)). We only need to consider x on the interval [0; 1]. Now, to find the sign of y', let's do following estimations: 1) 0≤x≤1 => 50≤50+x≤51 and 49≤50-x≤50; 2) consequently, ln49≤ln(50-x)≤ln50; 3) e log(3; 49) ln50
@wweraw48956 годин тому
Too easy let p=50^50 and q=49^51 Then take log(p) and log(q). Since logx function will always evaluate bigger positive numbers for x>1 you can easily find out which one is bigger. In this case log(q)>log(p)
@benji17754 місяці тому
glad to hear someone pronounce Euler as I did in the past lol
@bumpypants32412 місяці тому
That's the correct way to pronounce it 😅
@user-lu3sq6nh4c6 місяців тому
I feel... Between n^n and (n-1)^(n+2), It looks when n5, (n-1)^(n+2) is bigger. (n=natural number) But I don't know if it's true, or how to porve it.
@TheThaLime5 місяців тому
You can just see it by looking at the numbers, that's how exponentials work Just like you can intuitively see that 1045 is larger than 983.
@stephen57116 місяців тому
If we take binomials expansion, then easy confirm (1+1/49) n of firs two order is bigger than 1. 50/49 is also bigger than 1 , then any 1.xxxxxxxxx multiple
@costicaCJ2 місяці тому
ok, but why for all n there is (1 + 1/n) ^ n < 3 ? maybe for a given n an so but not for all. is the string ascended starting from 1 ? maybe yes but please confirm.
@ghanshyamborisagar68962 дні тому
Applying binomial theorem, ( 50-1) power 51, then expand then common 50 to power 50 is so simple
@marcelospaiva5 місяців тому
50^50 os 3.125 *10^12 e 49^51 is 1.6 * 10*16. De second one is 4 times greater than the first one.