If {(1 + x)^n} = {C_0} + {C_1}x + {C_2}{x^2} | vedclass

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Question

(b) We have, ${(1 + x)^n} = {C_0} + {C_1}x + {C_2}{x^2}.... + {C_n}{x^n}$

${\left( {1 + \frac{1}{x}} \right)^n} = {C_0} + {C_1}.\frac{1}{x} + {C_2}

Vedclass · Accepted Answer

$\frac{{(2n)!}}{{(n - 2)!(n + 2)!}}$

Vedclass · Answer

(b) We have, ${(1 + x)^n} = {C_0} + {C_1}x + {C_2}{x^2}.... + {C_n}{x^n}$

${\left( {1 + \frac{1}{x}} \right)^n} = {C_0} + {C_1}.\frac{1}{x} + {C_2}.\frac{1}{{{x^2}}} + ... + {C_n}\left( {\frac{1}{{{x^n}}}} \right)$

on multiplying both expansions, we get

$\frac{{{{(1 + x)}^{2n}}}}{{{x^n}}} = \sum {C_0^2 + x\sum {{C_0}{C_1} + {x^2}\sum {{C_0}{C_2} + ....} } } $$ + {x^r}\sum {{C_0}{C_r} + .....} $

The various sigma are the coefficient of ${x^0},x,{x^2},.....,{x^r}$ in $L.H.S.$ $\frac{{{{(1 + x)}^{2n}}}}{{{x^n}}}$ or coefficient of ${x^n},{x^{n + 1}},{x^{n + 2}},.....,{x^{n + r}}$ in the expansion of ${(1 + x)^{2n}}$ which occur in ${T_{n + 1,}}{T_{n + 2}},....$ and are

$^{2n}{C_n}{,^{2n}}{C_{n + 1}}{,^{2n}}{C_{n + 2}}{....^{2n}}{C_{n + r}}$etc.

$^{\,2n}{C_{n + 2}} = \frac{{(2\,n\,)\,!}}{{(n - 2)\,!\,(n + 2)\,!}}$

If ${(1 + x)^n} = {C_0} + {C_1}x + {C_2}{x^2} + .... + {C_n}{x^n}$, then ${C_0}{C_2} + {C_1}{C_3} + {C_2}{C_4} + {C_{n - 2}}{C_n}$ equals

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