JEE PYQ: Binomial Theorem Question 2
Question 2 - 2021 (16 Mar Shift 1)
Let $[x]$ denote greatest integer less than or equal to $x$. If for $n \in \mathbb{N}$, $(1 - x + x^3)^n = \sum_{j=0}^{3n} a_j x^j$, then $\sum_{j=0}^{\lfloor 3n/2 \rfloor} a_{2j} + 4 \sum_{j=0}^{\lfloor (3n-1)/2 \rfloor} a_{2j} + 1$ is equal to:
(1) 2 (2) $2^{n-1}$ (3) 1 (4) $n$
Show Answer
Answer: (3)
Solution
Put $x = 1$: sum of all $a_j = 1$. Put $x = -1$: alternating sum $= 1$. Adding gives $\sum a_{2j} = 1$.
BETA
