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 answers are presented.

The sequence shown below

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 \[ s_n = a_1 + a_2 + a_3 + ... a_n = a_1 \dfrac{1 - r^n}{1-r} \] The sum S of an infinite (n approaches infinity) geometric sequence and when |r| < 1 is given by

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

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
\(
a_2 = a_1 \times r = 10 (-1) = - 10 \\
a_3 = a_2 \times r = - 10 (-1) = 10 \\
a_4 = a_3 \times r = 10 (-1) = - 10 \\
a_5 = a_4 \times r = - 10 (-1) = 10
\)

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
\[
a_n = a_1 \times r^{n-1}
\]
that gives the n th term to find a_{10} as follows

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

Problem 3

Find a_{20} of a geometric sequence if the first few terms of the sequence are given by

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

\( a_20 = a_1 \times r^{20-1} = (-1/2)\times (-1/2)^{19} = (-1/2)^{20} = 1 / 2^{20} \)

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

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

We now divide the terms a_{10} and a_{15} to write

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

Simplify expressions in the above equation to obtain.

r^{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

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

__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 a geometric 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

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

__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

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.

a) a_{20} = 2^{18}

b) a_{30} = 1 / 2^{28}

c) r = 0.1

d) 0.9717171... = 481/495

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