Name: Sum of Infinite Terms of G.P.
Uploaded: 2019-07-19
Duration: 3 min 36 s
Description: Do you know that we can find the sum of an infinite number of terms of G.P.? Of course, there will be some conditions. Check this video to learn more.

Question

Find the sum of

n

terms of the series the

r th

term of which is

(2 r + 1) 2^{r}

.

Accepted Answer

Given, $r th$ term $(T_{r}) = (2 r + 1) . 2^{r}$

Putting $r = 1, 2, 3, 4, \dots \dots, n$ ,we get:

$T_{1} = 3 . 2^{1}$

$T_{2} = 5 . 2^{2}$

$T_{3} = 7 . 2^{3}$

$T_{4} = 9 . 2^{4}$

$T_{n} = (2 n + 1) . 2^{n}$

Now, the sum of series is:

$S = 3 . 2^{1} + 5 . 2^{2} + 7 . 2^{3} + 9 . 2^{4} + \dots \dots + (2 n + 1) . 2^{n}$

We observe that each term in the sum of the series is the product of the terms of an $A . P .$ and a $G . P .$

Thus, the series is an Arithmetico-Geometric Progression $(A . G . P .)$ .

This sum of series can be solved by multiplying both sides by $2$ (common ratio of the $G . P .$ ) and subtracting it from the original sum of series,

$∴ 2 . S = 0 + 3 . 2^{2} + 5 . 2^{3} + 7 . 2^{4} + \dots + (2 n - 1) . 2^{n} + (2 n + 1) . 2^{n + 1}$

On subtracting we get,

$S - 2 S = 3 . 2^{1} + 2 . 2^{2} + 2 . 2^{3} + \dots + 2 . 2^{n} - (2 n + 1) . 2^{n + 1} \Rightarrow - S = 6 + 2 . 2^{2} + 2 . 2^{3} + \dots + 2 . 2^{n} - (2 n + 1) . 2^{n + 1} \Rightarrow - S = 2^{1} + 2^{2} + 2^{3} + \dots + 2^{n + 1} - (2 n + 1) . 2^{n + 1} \Rightarrow - S = \frac{2 (2^{n + 1} - 1)}{1} - (2 n + 1) . 2^{n + 1} \Rightarrow - S = 2^{n + 2} - 2 - 2^{n + 2} . n - 2^{n + 1} \Rightarrow - S = 2^{n + 1} . 2 - 2 - 2^{n + 2} . n - 2^{n + 1} \Rightarrow - S = 2^{n + 1} - 2 - 2^{n + 2} . n \Rightarrow S = - 2^{n + 1} + 2 + 2^{n + 2} . n$

$∴$ $S = n . 2^{n + 2} - 2^{n + 1} + 2$

Find the sum of $n$ terms of the series the $r th$ term of which is $(2 r + 1) 2^{r}$ .

Important Questions on Sequences and Series

Learn and Prepare for any exam you want !

Find the sum of n terms of the series the rth term of which is (2r+1)2r.

Important Questions on Sequences and Series

Learn and Prepare for any exam you want !

Find the sum of $n$ terms of the series the $r th$ term of which is $(2 r + 1) 2^{r}$ .