Geometric sequences are used in several branches of applied mathematics to engineering, sciences, computer sciences, biology, finance...

Problems and exercises involving geometric sequences, along with detailed solutions and answers, are presented.

__REVIEW OF GEOMETRIC SEQUENCES__

The sequence shown below

8 = 2 \times 4 \\
32 = 8 \times 4 \\
128 = 32 \times 4 \\
\text{and so on}

The terms in the sequence may also be written as follows

a_1 = 2 \\
a_2 = a_1 \times 4 = 2 \times 4 \\
a_3 = a_2 \times 4 = 2 \times 4^2 \\
a_4 = a_3 \times 4 = 2 \times 4^3 \\

The n th term may now be written as

a_n = a_1 r^{n-1}

where a

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 \dfrac{1 - r^n}{1-r}

S = \dfrac{a_1}{1-r}

Arithmetic Series Online Calculator . An online calculator to calculate the sum of the terms in an arithmetic sequence.

__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 \times r = 10 (-1) = - 10 \\
a_3 = a_2 \times r = - 10 (-1) = 10 \\
a_4 = a_3 \times r = 10 (-1) = - 10 \\
a_5 = a_4 \times r = - 10 (-1) = 10

Find the 10 th term of a geometric sequence if a

a_n = a_1 \times r^{n-1}

a_{10} = 45 \times 0.2^{10-1} = 2.304 \times 10^{-5}

Find a

We first use the first few terms to find the common ratio r

r = a_2 / a_1 = (1/4) / (-1/2) = -1/2 \\
\\
r = a_3 / a_2 = (-1/8) / (1/4) = -1/2 \\

The common ration r = -1/2. We now use the formula a
a_20 = a_1 \times r^{20-1} = (-1/2)\times (-1/2)^{19} = (-1/2)^{20} = 1 / 2^{20}

Given the terms a

a_{10} = a_1 \times r^{10-1} = a_1 r^9 = 3 / 512 \\
\\
a_{15} = a_1 \times r^{15-1} = a_1 r^{14} = 3 / 16384

We now divide the terms a
a_{15} / a_{10} = a_1 \times r^{14} / (a_1 \times r^9) = (3 / 16384) / (3 / 512)

Simplify expressions in the above equation to obtain. r

We now use a

a_{10} = 3 / 512 = a_1 (1/2)^9

Solve for a
a_1 = 3

We now use the formula for the n th term to find a
a_{30} = 3(1/2)^{29} = 3 / 536870912

Find the sum

S = \sum_{k=1}^{6} 3^{k - 1}

S = 1 + 3 + 9 + 27 + 81 + 243 = 364

Another method is to first note that the terms making the sum are those of a geometric sequence with a

s

Find the sum

S = \sum_{i=1}^{10} 8 \times (1/4)^{i - 1}

8 , 8× ((1/4)

These are the terms of a geometric sequence with a

s

= 8 × (1 - (1/4)

Write the rational number 5.31313131... as the ratio of two integers.

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

S = a

We now write 5.313131... as follows

5.313131... = 5 + 31/99 = 526 / 99

Answer the following questions related to geometric sequences:

a) Find a

b) Find a

c) Find r given that a

d) write the rational number 0.9717171... as a ratio of two positive integers.

a) a

b) a

c) r = 0.1

d) 0.9717171... = 481/495

More math problems with detailed solutions in this site.

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Updated: 28 July 2018 (A Dendane)