[JEE Advanced 1993] Prove that $\sum_{r=1}^{k}(-3)^{r-1}\binom{3n}{2r-1}=0$, where $k=\frac{3n}2$ and $n$ is even.
Prove that $\sum_{r=1}^{k}(-3)^{r-1}\binom{3n}{2r-1}=0$, where $k=\frac{3n}2$ and $n$ is even.
1 Answer
VAVidaara Admin
✓ Vidaara Team
✓ Accepted
· 1mo ago
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Answer: Proved.
It is the imaginary part of $(1+i\sqrt3)^{3n}=2^{3n}\big(\cos\frac{3n\pi}3+i\sin n\pi\big)$; since $\sin n\pi=0$ for integer $n$, the alternating odd-index sum vanishes.
JEE Advanced 1993 · Binomial Theorem — verified solution by the Vidaara Team.
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