prove that:

if
(1+2x+2x²)ⁿ=a₀+a₁x+a₂x²+........a_2n²ⁿ
then prove that :
a_r=ⁿC₀.²ⁿC_r + ⁿC₁.^2n-2C_r-2 +ⁿC₂.^2n-4C_r-4+.....................

2 Answers

Had 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.....

341

Hari Shankar ·2009-11-27 01:23:11

Write 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 x^r 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.

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