What is Geometric Sequence

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: \(96, \ 48, \ 24, \ 12, \ 6, \ ...\)

 

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: \(96, \ 48, \ 24, \ 12, \ 6, \ ...\)

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

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

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

Geometric Sequences Practice Quiz