The series associated with a **geometric sequenc**e is known as a
**geometric series**. For example, 2 + 4 + 8 + 16 + 32 is the geometric series
associated with the geometric sequence 2, 4, 8, 16, 32.

Let a be the first term, r be the common ratio, n be the number
of terms, l be the last term and S_{n} be the sum to n terms of GS, then

S_{n} = a + ar + ar^{2} + … … … + ar^{n-2}
+ ar^{n-1} ………. (i)

Multiplying both sides by r,

rS_{n} = ar + ar^{2} + ar^{3} + … … … +
ar^{n-1} + ar^{n} …….. (ii)

Subtracting (i) from (ii), we get

(r – 1)S_{n} = -a + ar^{n}

^{}

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**Worked Out Example**

**Example 1:** Find the sum of the geometric series 1 + 2 + 4 + … … … 7 terms.

**Solution:**

Here,

First term (a) = 1

Common ratio (r) = 2/1 = 2

Number of terms (n) = 7

Sum of the terms (S_{n}) = ?

By using formula,

∴ S_{n} = 127

**Example 2:** In a GP, the first term is 7, the last term is 448 and the sum is
889, find the common ratio.

**Solution:**

Here,

First term (a) = 7

Last term (l) = 448

Sum of terms (S_{n}) = 889

Common ratio (r) = ?

By using the formula,

∴ The common ratio is 2.

**Solution:**

Here,

^{1+1}+ 3

^{2+1}+ 3

^{3+1}+ 3

^{4+1}+ 3

^{5+1}

= 3^{2} + 3^{3} + 3^{4} + 3^{5}
+ 3^{6}

= 9 + 27 + 81 + 243 + 729

= 1089

**Example 4:** How many terms of the series 32 + 48 + 72 + … … will add upto 665?

**Solution:**

Here,

First term (a) = 32

Common ratio (r) = 48/32 = 3/2

Sum of terms (S_{n}) = 665

Number of terms (n) = ?

By using formula,

∴ n = 6

∴ The number of terms = 6

**Example 5:** If S_{3} and S_{6} of a GS are 7 and 63
respectively, find the common ratio.

**Solution:**

Here,

^{3}+ 1 = 9

or, r^{3} = 9 – 1

or, r^{3} = 8

or, r^{3} = 2^{3}

or, r = 2

∴ Common ratio = 2

**Example 6:** The second and fifth terms of GS are 3 and 81 respectively. Find the
sum of the first five terms.

**Solution:**

Here,

The second term of GS is 3

i.e. t_{2} = 3

or, ar = 3

Again, the fifth term of GS is 81

i.e. t_{5} = 81

or, ar^{4} = 81

or, ar.r^{3} = 81

or, 3.r^{3} = 81

or, r^{3} = 81/3

or, r^{3} = 27 = 3^{3}

or, r = 3

And, ar = 3

or, a×3 = 3

or, a = 1

Now,

**Example 7:** Find the GP for which the sum of the first two terms is -4 and the
fifth term is 4 times the third term.

**Solution:**

Here,

S_{2} = -4

i.e. t_{1} + t_{2} = -4

or, a + ar = -4

or, a(1 + r) = -4 ……………… (i)

t_{5} = 4 × t_{3}

i.e. ar^{4} = 4 × ar^{2}

or, r^{2} = 4

or, r = ±2

Taking r = 2, from equation (i), we have

a(1 + 2) = -4

or, a = -4/3

The GP is a, ar, ar^{2}, …

i.e. -4/3, -4×2/3, -4×22/3, …

i.e. -4/3, -8/3, -16/3, …

Taking r = -2, from equation (i), we have

a(1 – 2) = -4

or, a = 4

The GP is a, ar, ar^{2}, …

i.e. 4, 4×-2, 4×(-2)^{2}, …

i.e. 4, -8, 16, …

**Example 8:** If the sum of the first three terms of a GP is 1 and the sum of the
first six terms is 28. Find the sum of the first 9 terms of the series.

**Solution:**

Here,

^{3}+ 1 = 28

or, r^{3} = 28 – 1

or, r^{3} = 27

or, r^{3} = 3^{3}

or, r = 3

∴ The sum of first 9 terms is 757.

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