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 required sum = displaystyle ∑ r = 1n(r2 + 1/r (r + 1))2r - 1= displaystyle ∑ r = 1n((2 r/r + 1) - (r - 1/r))2r - 1 = displaystyle ∑ r = 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

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

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

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

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

The required sum
$= r = 1 \sum n \frac{r ^{2} + 1}{r ( r + 1 )} 2^{r - 1} = r = 1 \sum n (\frac{2 r}{r + 1} - \frac{r - 1}{r}) 2^{r - 1}$
$= r = 1 \sum n (\frac{r}{r + 1} 2^{r} - \frac{r - 1}{r} 2^{r - 1}) = \frac{n}{n + 1} \cdot 2^{n}$

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