Let (1+x+2x^2)^{20} = a_0 + a_1x + a_2x^2 + l | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Given $(1+x+2x^2)^{20} = \sum_{k=0}^{40} a_k x^k$.Let $f(x) = (1+x+2x^2)^{20}$.$f(1) = a_0 + a_1 + a_2 + \ldots + a_{40} = (1+1+2)^{20} = 4^{20} =

Vedclass · Accepted Answer

$2^{19}(2^{20}-21)$

Vedclass · Answer

(B) Given $(1+x+2x^2)^{20} = \sum_{k=0}^{40} a_k x^k$.Let $f(x) = (1+x+2x^2)^{20}$.$f(1) = a_0 + a_1 + a_2 + \ldots + a_{40} = (1+1+2)^{20} = 4^{20} = 2^{40}$.$f(-1) = a_0 - a_1 + a_2 - \ldots + a_{40} = (1-1+2)^{20} = 2^{20}$.Subtracting the two equations: $f(1) - f(-1) = 2(a_1 + a_3 + \ldots + a_{39}) = 2^{40} - 2^{20}$.So,$a_1 + a_3 + \ldots + a_{39} = \frac{2^{40} - 2^{20}}{2} = 2^{39} - 2^{19}$.We need $a_1 + a_3 + \ldots + a_{37} = (a_1 + a_3 + \ldots + a_{39}) - a_{39}$.To find $a_{39}$,we look at the coefficient of $x^{39}$ in $(1+x+2x^2)^{20}$.Using the multinomial expansion,the term $x^{39}$ is obtained from $\frac{20!}{n_1! n_2! n_3!} (1)^{n_1} (x)^{n_2} (2x^2)^{n_3}$ where $n_1+n_2+n_3=20$ and $n_2+2n_3=39$.If $n_3=20$,$n_2=-1$ (impossible). If $n_3=19$,$n_2=1$,then $n_1=0$.Coefficient $a_{39} = \frac{20!}{0! 1! 19!} (1)^0 (1)^1 (2)^{19} = 20 	imes 2^{19}$.Thus,$a_1 + a_3 + \ldots + a_{37} = 2^{39} - 2^{19} - 20 	imes 2^{19} = 2^{39} - 21 	imes 2^{19} = 2^{19}(2^{20} - 21)$.

Let $(1+x+2x^2)^{20} = a_0 + a_1x + a_2x^2 + \ldots + a_{40}x^{40}$,then $a_1 + a_3 + a_5 + \ldots + a_{37}$ is equal to

Similar Questions

The coefficient of the middle term in the binomial expansion in powers of $x$ of $(1 + \alpha x)^4$ and of $(1 - \alpha x)^6$ is the same if $\alpha$ equals

If $f(n)$ is the coefficient of $x^n$ in the expansion of $(1+x)(1-x)^n$,then $f(2021)=$

Write the general term in the expansion of $(x^{2}-yx)^{12}, x \neq 0$.

If the sum of the coefficients of $x^r$ $(r=0, 1, 2, \ldots, 2n)$ in the expansion of $(1+3x-2x^2)^n$ is $128$,then $\sum_{r=1}^{2n} r \frac{^{2n}C_r}{^{2n}C_{r-1}} = $

The $r^{th}$ term in the expansion of $(a + 2x)^n$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module