11Had I been in ur place - trying to sort this horrible thing out, I'd have surely gone for induction, I feel that's the best for these sums.....
if
(1+2x+2x2)n=a0+a1x+a2x2+........a2n2n
then prove that :
ar=nC0.2nCr + nC1.2n-2Cr-2 +nC2.2n-4Cr-4+.....................
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2 Answers
11
341Write it as \left[x^2+(x+1)^2 \right]^n = \sum_{m=0}^n \binom {n}{m} x^{2m} (x+1)^{2(n-m)}
Then it is evident that the coefficient of xr will be
\sum_{m=0}^{\left[\frac{m}{2} \right]} \binom {n}{m} \binom {2n-2m}{r-2m} which is identical to the expression given.
