Easy Polynomo!

Let $n$ be a positive integer and $P(x)$ a polynomial of degree $2n$ such that $P(0) = 1$ and $P(k) = 2^{k-1}$ for $k=1,2,3, \ldots, 2n$ .

Prove that $2P(2n+1) - P(2n+2)=1$ .

#Algebra

Easy Math Editor

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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`	$\sin \theta$
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Comments

I found a proof with method of differences. Interesting how one can consider part of the chart to be a function following f(n)=2^n for n in domain, but then have an extra column with alternating 1s and 0s. My latex is bad. I hope you see where I am going.

Sal Gard - 4 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}`	$\boxed{123}$