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

2 n

terms of a series of which every even term is

a

times the term before it, and every odd term is

c

times the term before it, the first term being unity.

Accepted Answer

On keeping first term unity, even term $a$ times the term before it and odd term $c$ times the term before it, the sequence formed is: $1, a, a c, a^{2} c, a^{2} c^{2}, a^{3} c^{2}, \dots \dots \dots 2 n$ terms

We need to find, $S_{2 n} =$ $1 + a + a c + a^{2} c + a^{2} c^{2} + a^{3} c^{2} + \dots \dots \dots 2 n$ terms

Here, we can see ratios $(r)$ of consecutive odd terms are equal.

Thus, odd terms are in $G . P .$ with common ratio $r = a c$ .

Also, the ratios $(r)$ of consecutive even terms are equal.

Thus, even terms are also in $G . P .$ with common ratio $r = a c$ .

Thus, we can write,

$S_{2 n} =$ ( $1 + a c + a^{2} c^{2} + \dots \dots \dots n$ terms) $+$ ( $a + a^{2} c + a^{3} c^{2} + \dots \dots$ $n$ terms)

Now, applying the formula for the sum of $n$ terms in $G . P .$ $(S_{n}) = \frac{A (1 - r^{n})}{1 - r}$ (where $A$ is first term and $r$ is the common ratio of the $G . P .$ )

$S_{2 n} = \frac{1 (1 - a^{n} c^{n})}{1 - a c} + \frac{a (1 - a^{n} c^{n})}{1 - a c}$

$\Rightarrow S_{2 n} = \frac{(1 - a^{n} c^{n}) (1 + a)}{1 - a c}$

$∴ S_{2 n} = \frac{(a^{n} c^{n} - 1) (1 + a)}{(a c - 1)}$

Find the sum of $2 n$ terms of a series of which every even term is $a$ times the term before it, and every odd term is $c$ times the term before it, the first term being unity.

Important Questions on Geometrical Progression

Learn and Prepare for any exam you want !

Find the sum of 2n terms of a series of which every even term is a times the term before it, and every odd term is c times the term before it, the first term being unity.

Important Questions on Geometrical Progression

Learn and Prepare for any exam you want !

Find the sum of $2 n$ terms of a series of which every even term is $a$ times the term before it, and every odd term is $c$ times the term before it, the first term being unity.