Binomial Theorem: Level 3 Challenges:

I just looked at this problem in Binomial Theorem: Level 3 Challenges:

Which of the following numbers is larger? 101^50 or 100^50 +99^50. The answer is 101^50.

The interesting thing is that 101^2 is less than 100^2 + 99^2. and 101^3 is less than 100^3 + 99^3.

So my question is : What value of n satisfies the equation 101^n = 100^n + 99^n.

I make it just under 48.23.

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
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Comments

It is about 48.2275

X X - 2 years, 10 months ago

Thanks, I made it just under 48.23.

Mike Holden - 2 years, 10 months ago

The only way to solve this is via numerical methods.

Pi Han Goh - 2 years, 10 months ago

I used a calculator, trying high and low values of n until I found an approximate value then honed in on the true value.

Mike Holden - 2 years, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$