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

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

Question

(C) Given $(1+2x)^{20} = a_0 + a_1x + a_2x^2 + \dots + a_{20}x^{20}$.Putting $x=1$,we get $3^{20} = a_0 + a_1 + a_2 + \dots + a_{20} \dots (i)$.Puttin

Vedclass · Accepted Answer

$\frac{5 \cdot 3^{20}+1}{2}$

Vedclass · Answer

(C) Given $(1+2x)^{20} = a_0 + a_1x + a_2x^2 + \dots + a_{20}x^{20}$.Putting $x=1$,we get $3^{20} = a_0 + a_1 + a_2 + \dots + a_{20} \dots (i)$.Putting $x=-1$,we get $(-1)^{20} = a_0 - a_1 + a_2 - a_3 + \dots + a_{20}$,so $1 = a_0 - a_1 + a_2 - a_3 + \dots + a_{20} \dots (ii)$.Adding $(i)$ and $(ii)$,we get $a_0 + a_2 + a_4 + \dots + a_{20} = \frac{3^{20}+1}{2}$.Subtracting $(ii)$ from $(i)$,we get $a_1 + a_3 + a_5 + \dots + a_{19} = \frac{3^{20}-1}{2}$.We need to evaluate $S = 3a_0 + 2a_1 + 3a_2 + 2a_3 + \dots + 2a_{19} + 3a_{20}$.This can be written as $S = 3(a_0 + a_2 + \dots + a_{20}) + 2(a_1 + a_3 + \dots + a_{19})$.Substituting the values,$S = 3 \left( \frac{3^{20}+1}{2} \right) + 2 \left( \frac{3^{20}-1}{2} \right)$.$S = \frac{3 \cdot 3^{20} + 3 + 2 \cdot 3^{20} - 2}{2} = \frac{5 \cdot 3^{20} + 1}{2}$.

Let $(1+2x)^{20} = a_0 + a_1x + a_2x^2 + \dots + a_{20}x^{20}$. Then $3a_0 + 2a_1 + 3a_2 + 2a_3 + 3a_4 + 2a_5 + \dots + 2a_{19} + 3a_{20}$ equals

Similar Questions

$2 \cdot {}^{20}C_0 + 5 \cdot {}^{20}C_1 + 8 \cdot {}^{20}C_2 + 11 \cdot {}^{20}C_3 + \dots + 62 \cdot {}^{20}C_{20}$ is equal to

The value of $\frac{C_1}{2} + \frac{C_3}{4} + \frac{C_5}{6} + \dots$ is equal to

In the expansion of $(1+a)^{m+n},$ prove that the coefficients of $a^{m}$ and $a^{n}$ are equal.

Choose the correct option regarding the following statements:
$1$. $C_0+C_2+C_4+\ldots+C_n=2^{n-1}$,if $n$ is even
$2$. $C_1+C_3+C_5+\ldots+C_{n-1}=2^{n-1}$,if $n$ is even

If ${}^{20}C_{r}$ is the coefficient of $x^{r}$ in the expansion of $(1+x)^{20}$,then the value of $\sum_{r=0}^{20} r^{2} \cdot {}^{20}C_{r}$ is equal to:

Vedclass Test Series

Exam Paper Generator

Online Exam Module