MathDB

For a natural number

n

, let

P(x)

be the polynomial of

2n

−th degree such that:

P(0) = 1

and

P(k) = 2^{k-1}

for

k = 1, 2, . . . , 2n

. Prove that

2P(2n + 1) - P(2n + 2) = 1

. P.S. I tried to prove it by firstly expressing this polynomial using Lagrange interpolation but get bored of computations - it seems like it can be done this way, but I'd like to see more 'clever' solution. :)

Problem Statement