If {(1 + x)^n} = {C_0} + {C₁}x + {C₂}{x^2} + ... + {Cₙ}{x^n} , then value of {C_0} + {C₂} +

English

Gujarati

Hindi

7.Binomial Theorem

easy

If ${(1 + x)^n} = {C_0} + {C_1}x + {C_2}{x^2} + ... + {C_n}{x^n}$, then the value of ${C_0} + {C_2} + {C_4} + {C_6} + .....$ is

A

${2^{n - 1}}$

B

${2^{n - 1}}$

C

${2^n}$

D

${2^{n - 1}} - 1$

Solution

(a) ${(1 + x)^n} = {C_0} + {C_1}x + {C_2}{x^2} + …. + {C_n}{x^n}$

Putting $x = 1$, we get

$ \Rightarrow {2^n} = {C_0} + {C_1} + {C_2} + ….. + {C_n}$…..(i)

or ${C_1} + {C_2} + {C_3} + …. + {C_n} = {2^n} – 1$ $[\,{C_0} = {\,^n}{C_0} = 1]$

Again, putting $x = -1$, we get

$0 = {C_0} – {C_1} + {C_2} – {C_3} + ….$

or ${C_0} + {C_2} + {C_4} + ….$$ = {C_1} + {C_3} + {C_5} + ….$ $ i.e. A = B$

Also from $(i), A + B =2^n$ or $A = {2^{n – 1}} = B$

Hence, ${C_0} + {C_2} + {C_4} + …. = {2^{n – 1}}$.

Standard 11

Mathematics

Share

Similar Questions

What is the coefficient of $x^{100}$ in $(1 + x + x^2 + x^3 +…. + x^{100})^3$ ?

normal

If the sum of the coefficients of all the positive even powers of $x$ in the binomial expansion of $\left(2 x^{3}+\frac{3}{x}\right)^{10}$ is $5^{10}-\beta \cdot 3^{9}$, then $\beta$ is equal to

hard

(JEE MAIN-2022)

If ${}^{21}{C_1} + 3.{}^{21}{C_3} + 5.{}^{21}{C_5} + ……19{}^{21}{C_{19}} + 21.{}^{21}{C_{21}} = k$ Then number of prime factors of $k$ is

normal

Statement $-1$: $\mathop \sum \limits_{r = 0}^n \left( {r + 1} \right)\left( {\begin{array}{{20}{c}}n\\r\end{array}} \right) = \left( {n + 2} \right){2^{n – 1}}$

Statement $-2$:$\;\mathop \sum \limits_{r = 0}^n \left( {r + 1} \right)\left( {\begin{array}{{20}{c}}n\\r\end{array}} \right){x^r}\; = {\left( {1 + x} \right)^n} + nx{\left( {1 + x} \right)^{n – 1}}$

hard

(AIEEE-2008)

Let $\alpha=\sum_{k=0}^n\left(\frac{\left({ }^n C_k\right)^2}{k+1}\right)$ and $\beta=\sum_{k=0}^{n-1}\left(\frac{{ }^n C_k{ }^n C_{k+1}}{k+2}\right)$. If $5 \alpha=6 \beta$, then $n$ equals

normal

(JEE MAIN-2024)