The sum of the series (2/1⋅2)+(5/2⋅3)⋅2+(10/3⋅4)⋅22 +(17/4⋅5)⋅23 +..... to n terms is

Question

The sum of the series (2/1⋅2)+(5/2⋅3)⋅2+(10/3⋅4)⋅22 +(17/4⋅5)⋅23 +..... to n terms is (A) ((n/n+1))2n+1 (B) ((n+1/n))2n+1 (C) ((n/n+1))2n (D) ((n+1/n))2n

Tardigrade Admin · Accepted Answer

The given series is ∑ limitsr=1n (r2+1/r(r+1)) 2r-1 = ∑ limitsr=1n ((2r/r+1) - (r-1/r))2r-1 =∑ limitsr=1n ((r/r+1)2r -(r-1/r)2r-1) = (n/ n+1) 2n

Q. The sum of the series
$\frac{2}{1 \cdot 2} + \frac{5}{2 \cdot 3} \cdot 2 + \frac{10}{3 \cdot 4} \cdot 2^{2} + \frac{17}{4 \cdot 5} \cdot 2^{3} + .....$ to $n$ terms is

$(\frac{n}{n + 1}) 2^{n + 1}$

$(\frac{n + 1}{n}) 2^{n + 1}$

$(\frac{n}{n + 1}) 2^{n}$

$(\frac{n + 1}{n}) 2^{n}$

The given series is
$r = 1 \sum n \frac{r ^{2} + 1}{r ( r + 1 )} 2^{r - 1} = r = 1 \sum n (\frac{2 r}{r + 1} - \frac{r - 1}{r}) 2^{r - 1}$
$= r = 1 \sum n (\frac{r}{r + 1} 2^{r} - \frac{r - 1}{r} 2^{r - 1}) = \frac{n}{n + 1} 2^{n}$

Q. The sum of the series 1⋅22​+2⋅35​⋅2+3⋅410​⋅22+4⋅517​⋅23+..... to n terms is

(n+1n​)2n+1

(nn+1​)2n+1

(n+1n​)2n

(nn+1​)2n