Question

If T=${}^{2n+1}{C}_1{+}^{2n+1}{C}_2+......{+}^{2n+1}{C}_n=255$. Find the value of n.

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Answer 1

In expansion of binomial expression
Coefficient of first term of binomial expansion=Coefficient of last term of binomial expansion
Similarly,
Coefficient of second term of binomial expansion= Coefficient of second last term of binomial expansion
And so on.....
=${}^{2n+1}{C}_0+{}^{2n+1}{C}_1+{}^{2n+1}{C}_2+......+{}^{2n+1}{C}_n+{}^{2n+1}{C}_{n+1}+{}^{2n+1}{C}_{n+2}+.....+{}^{2n+1}{C}_{2n+1}=(1+1{)}^{2n+1}=2^{2n+1}$
$\Rightarrow {}^{2n+1}{C}_0+2({}^{2n+1}{C}_1+{}^{2n+1}{C}_2+......+{}^{2n+1}{C}_n)+{}^{2n+1}{C}_{2n+1}=2^{2n+1}$
$\Rightarrow 1+2\left(T\right)+1=2^{2n+1}\Rightarrow 1+T=\frac{2^{2n+1}}{2}=2^{2n}$
$\Rightarrow 1+255=2^{2n}\Rightarrow 2^{2n}=2^8\Rightarrow n=4$