Math CentralQuandaries & Queries


Question from Anuj, a student:

if (1+x+xsquare) whole raise to 20 is a0 + a1x + ........ + a20xraise to 40, then find a0 + a2 + a4 + .......... + a38.
How do you solve this???????

First off, that last coefficient should be a_40. terms; your expanded expression will have the form

$a_0 + a_1x + ... a_{40}x^{40}.$

Look at the more general sums of even and odd coefficients
of $(1+x+x^2)^n$

1 (E(0) = 1, O(0)=0)
1+1+1 (E(1) = 2, O(1)=1)
1+2+3+2+1 (E(2) = 5, O(2)=4)
1+3+6+7+6+3+1 (E(3) = 14, O(3)=13)

You should be able to find a conjecture easily (look at E(n)+O(n) if you're stuck)

Prove it by induction: assume it true for (E(n),O(n)) and use
that to prove it for (E(n+1),O(n+1)).

Good Hunting!

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