Geometric Sequences Problems with Solutions
Problems and exercises involving geometric sequences, along with detailed solutions and answers, are presented.
REVIEW OF GEOMETRIC SEQUENCES
The sequence shown below
2 , 8 , 32 , 128 , ...
has been obtained starting from 2 and multiplying each term by 4. 2 is the first term of the sequence and 4 is the common ratio.
8 = 2 * 4 or 8 / 2 = 4
32 = 8 * 4 or 32 / 8 = 4
128 = 32 * 4 or 128 / 32 = 4 and so on.
The terms in the sequence may also be written as follows
a_{1} = 2
a_{2} = a_{1} * 4 = 2 * 4
a_{3} = a_{2} * 4 = 2 * 4^{2}
a_{4} = a_{3} * 4 = 2 * 4^{3}
The n th term may now be written as
a_{n} = a_{1} * r^{n1}
where a_{1} is the first term of the sequence and r is the common ratio which is equal to 4 in the above example.
The sum of the first n terms of a geometric sequence is given by
s_{n} = a_{1} + a_{2} + a_{3} + ... a_{n} = a_{1} (1  r^{n}) / (1  r)
The sum S of an infinite (n approaches infinity) geometric sequence and when r <1 is given by
S = a_{1} / (1  r)
Problem 1:
Find the terms a_{2}, a_{3}, a_{4} and a_{5} of a geometric sequence if a_{1} = 10 and the common ratio r =  1.
Solution to Problem 1:

Use the definition of a geometric sequence
a_{2} = a_{1} * r = 10 * (1) =  10
a_{3} = a_{2} * r =  10 * (1) = 10
a_{4} = a_{3} * r = 10 * (1) =  10
a_{5} = a_{4} * r =  10 * (1) = 10
Problem 2:
Find the 10 th term of a geometric sequence if a_{1} = 45 and the common ration r = 0.2.
Solution to Problem 2:

Use the formula a_{n} = a_{1} * r^{n1} that gives the n th term to find a_{10} as follows
a_{10} = a_{1} * r^{n1}
= 45 * 0.2^{9} = 2.304 * 10^{5}
Problem 3:
Find a_{20} of a geometric sequence if the first few terms of the sequence are given by
1/2 , 1/4 , 1/8 , 1 / 16 , ...
Solution to Problem 3:

We first use the first few terms to find the common ratio
r = a_{2} / a_{1} = (1/4) / (1/2) = 1/2
r = a_{3} / a_{2} = (1/8) / (1/4) = 1/2
r = a_{4} / a_{3} = (1/16) / (1/8) = 1/2

The common ration r = 1/2. We now use the formula a_{n} = a_{1} * r^{n1} for the n th term to find a_{20} as follows.
a_{20} = a_{1} * r^{201}
= (1/2) * (1/2)^{201} = 1 / (20^{20})
Problem 4:
Given the terms a_{10} = 3 / 512 and a_{15} = 3 / 16384 of a geometric sequence, find the exact value of the term a_{30} of the sequence.
Solution to Problem 4:

We first use the formula for the n th term to write a_{10} and a_{15} as follows
a_{10} = a_{1} * r^{101} = 3 / 512
a_{15} = a_{1} * r^{151} = 3 / 16384

We now divide the terms a_{10} and a_{15} to write
a_{15} / a_{10} = (a_{1} * r^{14} / a_{1} * r^{9}) = (3 / 16384) / (3 / 512)

Solve for r to obtain.
r^{5} = 1 / 32 which gives r = 1/2

We now use a_{10} to find a_{1} as follows.
a_{10} = 3 / 512 = a_{1}(1/2)^{9}

Solve for a_{1} to obtain.
a_{1} = 3

We now use the formula for the n th term to find a_{30} as follows.
a_{30} = 3(1/2)^{29} = 3 / 536870912
Problem 5:
Find the sum
Solution to Problem 5:

We first rewrite the sum S as follows
S = 1 + 3 + 9 + 27 + 81 + 243 = 364

Another method is to first note that the terms making the sum are those of an arithmetic sequence with a_{1} = 1 and r = 3 using the formula s_{n} = a_{1} (1  r^{n}) / (1  r) with n = 6.
s_{6} = 1 (1  3^{6}) / (1  3) = 364
Problem 6:
Find the sum
10 
S = ∑ 8*(1/4)^{i1}

i=1 
Solution to Problem 6:

An examination of the terms included in the sum are
8 , 8*((1/4)^{1} , 8*((1/4)^{2} , ... , 8*((1/4)^{9}

These are the terms of a geometric sequence with a_{1} = 8 and r = 1/4 and therefore we can use the formula for the sum of the terms of a geometric sequence
s_{10} = a_{1} (1  r^{n}) / (1  r)
= 8 * (1  (1/4)^{10}) / (1  1/4) = 10.67 (rounded to 2 decimal places)
Problem 7:
Write the rational number 5.31313131... as the ratio of two integers.
Solution to Problem 7:

We first write the given rational number as an infinite sum as follows
5.313131... = 5 + 0.31 + 0.0031 + 0.000031 + ....

The terms making 0.31 + 0.0031 + 0.000031 ... are those of a geometric sequence with a_{1} = 0.31 and r = 0.01. Hence the use of the formula for an infinite sum of a geometric sequence
S = a_{1} / (1  r) = 0.31 / (1  0.01) = 0.31 / 0.99 = 31 / 99

We now write 5.313131... as follows
5.313131... = 5 + 31/99 = 526 / 99
Exercises:
Answer the following questions related to geometric sequences:
a) Find a_{20} given that a_{3} = 1/2 and a_{5} = 8
b) Find a_{30} given that the first few terms of a geometric sequence are given by 2 , 1 , 1/2 , 1/4 ...
c) Find r given that a_{1} = 10 and a_{20} = 10^{18}
d) write the rational number 0.9717171... as a ratio of two positive integers.
Solutions to Above Exercises:
a) a_{20} = 2^{18}
b) a_{30} = 1 / 2^{28}
c) r = 0.1
d) 0.9717171... = 481/495
More math problems with detailed solutions in this site.
Arithmetic Series Online Calculator. An online calculator to calculate the sum of the terms in an arithmetic sequence.
 
Home Page 
HTML5 Math Applets for Mobile Learning 
Math Formulas for Mobile Learning 
Algebra Questions  Math Worksheets

Free Compass Math tests Practice
Free Practice for SAT, ACT Math tests

GRE practice

GMAT practice
Precalculus Tutorials 
Precalculus Questions and Problems

Precalculus Applets 
Equations, Systems and Inequalities

Online Calculators 
Graphing 
Trigonometry 
Trigonometry Worsheets

Geometry Tutorials 
Geometry Calculators 
Geometry Worksheets

Calculus Tutorials 
Calculus Questions 
Calculus Worksheets

Applied Math 
Antennas 
Math Software 
Elementary Statistics
High School Math 
Middle School Math 
Primary Math
Math Videos From Analyzemath
Author 
email
Updated: 2 April 2013
Copyright © 2003  2014  All rights reserved