What is Geometric Sequence

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Geometric Sequences

Geometric sequences are those in which the following number is obtained by multiplying the previous term by a constant known as the common ratio. The letter $r$ is used to show the common ratio.

We have an increasing geometric sequence if $r$ is more than $1$ . We have a decreasing geometric sequence if $r$ is between $0$ and $1$ .

The Geometric Sequences Formula

We can use the geometric sequence formula to find any number in the sequence.

$a_n = a \ r^{(n-1)}$ , where:

$a$ : The first term
$a_n$ : The $n$ th term
$n$ : The term poistion
$r$ : Common ratio

Finding the terms in a Geometric Sequence

If we divide any term by the previous term, we can find the common ratio: $r \ = \ \frac{a_n}{a_{n \ - 1}}$

Example

Consider the geometric sequence: $2, \ 6, \ 18, \ 54, \ 162, \ ...$ , Find $a_{7} \ = \ ?$

Solution

Before using the geometric sequence formula, we need to know the first term, the common ratio, and the position of the term we want to find:

The first term: $a \ = \ 2$
common ratio: $r \ = \ 3$
Term's position: $n \ = \ 7$

Now, we fill in the formula with these numbers:

$a_n = a \ r^{(n-1)} \ ⇒ \ a_7 = 2 \ 3^{(6)} \ = \ 2 \times 729 \ = \ 1458$

We can see that the number is very big. Depending on what the common ratio is, geometric sequences tend to grow quickly.

The formula for the Sum of a Geometric Sequence

With the sum of a finite geometric sequence formula, you can find the sum of a geometric sequence's first $n$ terms. Consider a geometric series with $n$ terms, where the first term is $a$ , and the ratio $r$ : $a, \ ar, \ ar^2, \ ar^3, \ ... \ , \ ar^{n \ - \ 1}$

The sum is shown by $S_n$ and given by the formula:

$S_n \ = \ \frac{a(r^n \ - \ 1)}{r \ - \ 1}$ , when $r \ ≠ \ 1$
$S_n \ = \ na$ , when $r \ = \ 1$

Example

Add up the first four terms in: $10, \ 30, \ 90, \ 270, \ 810, \ 2430, \ ...$

Solution

The first term: $a \ = \ 10$
common ratio: $r \ = \ 3$
$n \ = \ 4$

So:

$S_n \ = \ \frac{a(r^n \ - \ 1)}{r \ - \ 1} \ = \ \frac{10(3^n \ - \ 1)}{3 \ - \ 1} \ = \ 400$

Free printable Worksheets

Exercises for Geometric Sequences

1) Find the explicit formula: $-1, \ 2, \ -4, \ 8, \ -16, \ ...$

2) Find the explicit formula: $-2, \ -6, \ -18, \ -54, \ -162, \ ...$

3) Find the explicit formula and $a_{11}$ : $88, \ 44, \ 22, \ 11, \ 5.5, \ ...$

4) Find the explicit formula and $a_5$ : $a \ = \ 5, \ r \ = \ -5$

5) Find the explicit formula and the first $7$ terms: $a \ = \ 2, \ r \ = \ \frac{1}{3}$

6) Find the first $10$ terms: $a \ = \ 7, \ r \ = \ -\frac{1}{2}$

7) Find $a_{7}$ : $45, \ , \ 9, \ \frac{9}{5} \ , \ \frac{9}{25} \ , \ ...$

8) Find $a_{10}$ : $a_7 \ = \ \frac{2}{27} \ , \ r \ = \ -\frac{1}{3}$

9) Find the explicit formula and the sum of the first five terms: $a \ = \ 10, \ r \ = \ 2$

10) Find the explicit formula and the sum of the first $10$ terms: $96, \ 48, \ 24, \ 12, \ 6, \ ...$

Answers

1) Find the explicit formula: $-1, \ 2, \ -4, \ 8, \ -16, \ ...$

$\color{red}{a \ = \ -1, \ r \ = \ \frac{2}{-1} \ = \ -2, \ a_n = a \ r^{(n-1)}}$
$\color{red}{⇒ \ a_n = -1 \ (-2)^{(n-1)}}$

2) Find the explicit formula: $-2, \ -6, \ -18, \ -54, \ -162, \ ...$

$\color{red}{a \ = \ -2, \ r \ = \ \frac{-6}{-2} \ = \ 3, \ a_n = a \ r^{(n-1)}}$
$\color{red}{⇒ \ a_n = -2 \ (3)^{(n-1)}}$

3) Find the explicit formula and $a_{11}$ : $88, \ 44, \ 22, \ 11, \ 5.5, \ ...$

$\color{red}{a \ = \ 88, \ r \ = \ \frac{44}{88} \ = \ \frac{1}{2}, \ a_n = a \ r^{(n-1)}}$
$\color{red}{⇒ \ a_n = 88 \ (\frac{1}{2} \ )^{(n-1)} \ ⇒ \ a_{11} = 88 \ (\frac{1}{2} \ )^{(11-1)} \ = \ \frac{88}{1024} \ = \ \frac{11}{256}}$

4) Find the explicit formula and $a_5$ : $a \ = \ 5, \ r \ = \ -5$

$\color{red}{a \ = \ 5, \ r \ = \ -5, \ a_n = a \ r^{(n-1)}}$
$\color{red}{⇒ \ a_n = 5 \ (-5)^{(n-1)} \ ⇒ \ a_5 = 5 \ (-5)^{(5-1)} \ = \ 5 \times \frac{1}{(-5)^4} \ = \ \frac{1}{125}}$

5) Find the explicit formula and the first $7$ terms: $a \ = \ 2, \ r \ = \ \frac{1}{3}$

$\color{red}{a \ = \ 2, \ r \ = \ \frac{1}{3} \ , \ a_n = a \ r^{(n-1)}}$
$\color{red}{⇒ \ a_n = 2 \ (\frac{1}{3} \ )^{(n-1)}}$
$\color{red}{⇒ 2, \ \frac{2}{3} \ , \ \frac{2}{9} \ , \ \frac{2}{27} \ , \ \frac{2}{81} \ , \ \frac{2}{243} \ , \ \frac{2}{729} \ , \ ...}$

6) Find the first $10$ terms: $a \ = \ 7, \ r \ = \ -\frac{1}{2}$

$\color{red}{a \ = \ 7, \ r \ = \ -\frac{1}{2} \ , \ a_n = a \ r^{(n-1)}}$
$\color{red}{⇒ \ a_n = 7 \ (-\frac{1}{2} \ )^{(n-1)}}$
$\color{red}{⇒ 7, \ -\frac{7}{2} \ , \ \frac{7}{4} \ , \ -\frac{7}{8} \ , \ \frac{7}{16} \ , \ -\frac{7}{32} \ , \ \frac{7}{64} \ , \ -\frac{7}{128} \ , \ \frac{7}{256} \ , \ -\frac{7}{512} \ , \ ...}$

7) Find $a_{7}$ : $45, \ , \ 9, \ \frac{9}{5} \ , \ \frac{9}{25} \ , \ ...$

$\color{red}{a \ = \ 45, \ r \ = \ \frac{1}{5}, \ a_n = a \ r^{(n-1)}}$
$\color{red}{⇒ \ a_n = 45 \ (\frac{1}{5} \ )^{(n-1)} \ ⇒ \ a_7 = 45 \ (\frac{1}{5} \ )^{(7-1)} \ = \ 45 \times \frac{1}{5^6} \ = \ \frac{9}{3125}}$

8) Find $a_{10}$ : $a_7 \ = \ \frac{2}{27} \ , \ r \ = \ -\frac{1}{3}$

$\color{red}{a_7 \ = \ \frac{2}{27} \ , \ r \ = \ -\frac{1}{3} \ , \ a_n = a \ r^{(n-1)}}$
$\color{red}{⇒ \ a_7 \ = \ a \ (-\frac{1}{3} \ )^{(7-1)} \ ⇒ \ a \ = \ \frac{a_7}{(-\frac{1}{3} \ )^6} \ = \ \frac{\frac{2}{27}}{\frac{1}{3^6}} \ = \ 54}$
$\color{red}{⇒ \ a_{10} = \ 54 \ (-\frac{1}{3} \ )^{(10-1)} \ = \ -\frac{2}{729}}$

9) Find the explicit formula and the sum of the first five terms: $a \ = \ 10, \ r \ = \ 2$

$\color{red}{a \ = \ 10, \ r \ = \ 2, \ a_n = a \ r^{(n-1)}}$
$\color{red}{⇒ \ a_n = 10 \ 2^{(n-1)}}$

$\color{red}{S_n \ = \ \frac{a(r^n \ - \ 1)}{r \ - \ 1} \ ⇒ \ S_5 \ = \ \frac{10(3^5 \ - \ 1)}{3 \ - \ 1} \ = \ 5(243 \ - \ 1) \ = \ 1210}$

10) Find the explicit formula and the sum of the first $10$ terms: $96, \ 48, \ 24, \ 12, \ 6, \ ...$

$\color{red}{a \ = \ 96, \ r \ = \ \frac{1}{2} \ , \ a_n = a \ r^{(n-1)}}$
$\color{red}{⇒ \ a_n = 96 \ (\frac{1}{2} \ )^{(n-1)}}$

$\color{red}{S_n \ = \ \frac{a(r^n \ - \ 1)}{r \ - \ 1}}$

$\color{red}{⇒ \ S_{10} \ = \ \frac{96((\frac{1}{2} \ )^{10} \ - \ 1)}{2 \ - \ 1} \ = \ \frac{96}{1023}}$

What is Geometric Sequence