Practice Problems on Geometric Series (original) (raw)

Last Updated : 23 Jul, 2025

A **geometric series is a type of infinite series formed by summing the terms of a geometric sequence. A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the **common ratio.

The general form of a geometric series can be expressed as:

S = a + ar + ar^2 + ar^3 + ar^4 + \ldots

Where:

Sample Questions of Geometric Series

**Question 1: What is the Geometric mean 2, 4, 8?

**Solution:

According to the formula,

=\sqrt [3]{(2)(4)(8)}\\=4

**Question 2: **Find the first term and common factor in the following Geometric Progression:

**4, 8, 16, 32, 64, . . .

**Solution:

Here, It is clear that the first term is 4, a=4

We obtain common Ratio by dividing 1st term from 2nd:

r = 8/4 = 2

**Question 3: **Find the 8 th and the n th **term for the G.P: 3, 9, 27, 81, . . .

**Solution:

Put n=8 for 8th term in the formula: arn-1

For the G.P : 3, 9, 27, 81 . . .

First term (a) = 3

Common Ratio (r) = 9/3 = 3

8th term = 3(3)8-1 = 3(3)7 = 6561

Nth = 3(3)n-1 = 3(3)n(3)-1

= 3n

**Question 4: **For the G.P. : 2, 8, 32, . . . which term will give the value 131073?

**Solution:

Assume that the value 131073 is the Nth term,

a = 2, r = 8/2 = 4

Nth term (an) = 2(4)n-1 = 131073

4n-1 = 131073/2 = 65536

4n-1 = 65536 = 48

Equating the Powers since the base is same:

n-1 = 8

n = 9

**Question 5: Find the sum up to 5 th and N th **term of the series: 1, \frac{1}{2},\frac{1}{4},\frac{1}{8}...

**Solution:

a= 1, r = 1/2

Sum of N terms for the G.P, {S_n =\frac{a(1-r^n)}{1-r}}

= \frac{1(1-(\frac{1}{2})^n)}{1-\frac{1}{2}}

= 2 (1-(\frac{1}{2})^n)

Sum of first 5 terms ⇒ a5 = 2 ( 1-(\frac{1}{2})^5)

**= 2 ( 1-(\frac{1}{32}))

**= (\frac{31}{16})

**Question 6: Find the Sum of the Infinite G.P: 0.5, 1, 2, 4, 8, ...

**Solution:

Formula for the Sum of Infinite G.P: \frac{a}{1-r} ; r≠0

a = 0.5, r = 2

S∞= (0.5)/(1-2) = 0.5/(-1)= -0.5

**Question 7: **Find the sum of the Series: 5, 55, 555, 5555,... n terms

**Solution:

The given Series is not in G.P but it can easily be converted into a G.P with some simple modifications.

Taking 5 common from the series: 5 (1, 11, 111, 1111,... n terms)

Dividing and Multiplying with 9: \frac{5}{9}(9+ 99+ 999+...n terms)

⇒ \frac{5}{9}[((10+(10)^2+(10)^3+...n terms)-(1+1+1+...n terms)]

⇒\frac{5}{9}[(\frac{10((10)^n-1)}{10-1})-(n)]

⇒ \frac{5}{9}[(\frac{10((10)^n-1)}{9})-(n)]

Worksheet: Geometric Series

Worksheet-on-Geometric-Series

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Download Free PDF Worksheet on Geometric Series

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