The value of frac{C_1}{2} + frac{C_3}{4} + fr | vedclass

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(A) We know that the expansion of binomial coefficients is given by:$(1 + x)^n = C_0 + C_1 x + C_2 x^2 + C_3 x^3 + C_4 x^4 + C_5 x^5 + \dots$$(1 - x)^

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$\frac{2^n - 1}{n + 1}$

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(A) We know that the expansion of binomial coefficients is given by:$(1 + x)^n = C_0 + C_1 x + C_2 x^2 + C_3 x^3 + C_4 x^4 + C_5 x^5 + \dots$$(1 - x)^n = C_0 - C_1 x + C_2 x^2 - C_3 x^3 + C_4 x^4 - C_5 x^5 + \dots$Subtracting the two equations:$(1 + x)^n - (1 - x)^n = 2(C_1 x + C_3 x^3 + C_5 x^5 + \dots)$Dividing by $2x$:$\frac{(1 + x)^n - (1 - x)^n}{2x} = C_1 + C_3 x^2 + C_5 x^4 + \dots$Integrating both sides from $x = 0$ to $x = 1$:$\int_0^1 \frac{(1 + x)^n - (1 - x)^n}{2x} dx = \int_0^1 (C_1 + C_3 x^2 + C_5 x^4 + \dots) dx$This approach is complex. Alternatively,using the property $\frac{C_k}{k+1} = \frac{1}{n+1} \binom{n+1}{k+1}$:$\sum_{k 	ext{ odd}} \frac{C_k}{k+1} = \frac{1}{n+1} \sum_{k 	ext{ odd}} \binom{n+1}{k+1}$Let $j = k+1$,then $j$ is even:$= \frac{1}{n+1} \sum_{j 	ext{ even}, j=2}^{n+1} \binom{n+1}{j} = \frac{1}{n+1} (2^{(n+1)-1} - 1) = \frac{2^n - 1}{n + 1}$

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

Similar Questions

Let $(1 + x)^m = C_0 + C_1x + C_2x^2 + C_3x^3 + . . . + C_mx^m$,where $C_r = {}^mC_r$ and $A = C_1C_3 + C_2C_4 + C_3C_5 + . . . + C_{m-2}C_m$. Which of the following is false?

If $(1 + x)^n = C_0 + C_1x + C_2x^2 + .......... + C_nx^n,$ then $C_0^2 + C_1^2 + C_2^2 + C_3^2 + ...... + C_n^2$ =

If $(\frac{1}{^{15}C_{0}}+\frac{1}{^{15}C_{1}})(\frac{1}{^{15}C_{1}}+\frac{1}{^{15}C_{2}})...(\frac{1}{^{15}C_{12}}+\frac{1}{^{15}C_{13}}) = \frac{a^{13}}{^{14}C_{0} \cdot ^{14}C_{1} \cdot ... \cdot ^{14}C_{12}}$, then $30a$ is equal to:

$3 \cdot C_0 + 7 \cdot C_1 + 11 \cdot C_2 + \ldots + (3 + 4n) C_n =$

For $2 \le r \le n$,$\binom{n}{r} + 2\binom{n}{r-1} + \binom{n}{r-2}$ is equal to

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