Let [x] denote greatest integer less than or equal to

Question:

Let $[x]$ denote greatest integer less than or equal to $x$. If for $n \in \mathbb{N},\left(1-x+x^{3}\right)^{n}=\sum_{j=0}^{3 n} a_{j} x^{j}$, then $\left.\sum_{j=0}^{\left[\frac{3 n}{2}\right]} a_{2 j}+4 \sum_{j=0}^{\left[\frac{3 n-1}{2}\right.}\right] a_{2 j}+1$ is equal to :

  1. (1) 2

  2. (2) $2^{n-1}$

  3. (3) 1

  4. (3) 1


Correct Option: , 3

Solution:

$\left(1-x+x^{3}\right)^{n}=\sum_{j=0}^{3 n} a_{j} x^{j}$

$\left(1-x+x^{3}\right)^{n}=a_{0}+a_{1} x+a_{2} x^{2} \ldots \ldots+a_{3 n} x^{3 n}$

$\sum_{j=0}^{\frac{3 n}{2}} a_{2 j}=\operatorname{Sum}$ of $a_{0}+a_{2}+a_{4} \ldots \ldots$

$\left[\frac{3 n-1}{2}\right]$

$\sum_{j=0} a_{2 j}+1=$ Sum of $a_{1}+a_{3}+a_{5} \ldots \ldots$

put $x=1$

$1=a_{0}+a_{1}+a_{2}+a_{3} \ldots \ldots \ldots+a_{3 n} \ldots \ldots(A)$

Put $x=-1$

$1=a_{0}-a_{1}+a_{2}-a_{3} \ldots \ldots \ldots+(-1)^{3 n} a_{3 n} \ldots \ldots$

Solving (A) and (B)

$a_{0}+a_{2}+a_{4} \ldots .=1$

$a_{1}+a_{3}+a_{5} \ldots \ldots=0$

$\left.\left.\sum_{j=0}^{\left[\frac{3 n}{2}\right.}\right] a_{2 j}+4 \sum_{j=0}^{\left[\frac{3 n-1}{2}\right.}\right] a_{2 j+1}=1$

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