24^25 or 25^24: Proving Which Is LargerDate: 03/12/2015 at 07:26:27 From: Gerry Subject: Inequality How would you prove that 24^25 > 25^24 without using a calculator? I thought of mathematical induction, but it's not the case that (a + 1)^a < a^(a + 1). Thank you. Date: 03/12/2015 at 08:37:02 From: Doctor Ali Subject: Re: Inequality Hi Gerry! Thanks for writing to Dr. Math. Let, A1 = 24^25 B1 = 25^24 To begin, let's raise both numbers to the power 1/(24*25). This gives A2 = 24^(1/24) B2 = 25^(1/25) Now, if we show that A2 is greater than B2, we can easily deduce that A1 is also greater than B1. To do this, we can consider y = x^(1/x) Taking the derivative of both sides gives (1 - ln(x)) x^(1/x) y' = --------------------- x^2 We can clearly see that y' is always a negative number when x is greater than e, which is about 2.718281. This means that this function is decreasing for x > e. And this in turn means that B2 < A2 Therefore, B1 < A1 So, 24^25 > 25^24 Does that make sense? For some similar problems, please take a look at the following pages from our archive: http://mathforum.org/library/drmath/view/70271.html http://mathforum.org/library/drmath/view/53916.html http://mathforum.org/library/drmath/view/61584.html I hope it helps. Please write back if you still have any difficulties. - Doctor Ali, The Math Forum http://mathforum.org/dr.math/ Date: 03/12/2015 at 09:00:37 From: Gerry Subject: Thank you (Inequality) Thank you! |
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