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Question: If \((1 + x)^{n}\) represent the terms in the expansion of \(2^{n} + 1\), then \(2^{n} - 1\) \(2^{n}...

If (1+x)n(1 + x)^{n} represent the terms in the expansion of 2n+12^{n} + 1, then 2n−12^{n} - 1 2n2^{n}

A

2n−12^{n - 1}

B

(x2+x−3)319(x^{2} + x - 3)^{319}

C

(x−2y+3z)n(x - 2y + 3z)^{n}

D

(1+x)n(1 + x)^{n}

Answer

(x2+x−3)319(x^{2} + x - 3)^{319}

Explanation

Solution

From the given condition, replacing a by ai and – ai respectively, we get

1.3 6mu.5...(2r−1)2r.(−1)r(−1)r2rr!\frac{1.3\mspace{6mu}.5...(2r - 1)}{2^{r}}.\frac{( - 1)^{r}( - 1)^{r}2^{r}}{r!} ....(i)

and 1.3.5...(2r−1)r!=2r!r!r!2r\frac{1.3.5...(2r - 1)}{r!} = \frac{2r!}{r!r!2^{r}} ......(ii)

Multiplying (ii) and (i) we get required result

i.e. ∑k=1nk(1+1n)k−1\sum_{k = 1}^{n}{k\left( 1 + \frac{1}{n} \right)^{k - 1}}