4.4 Geometric Sequence and Common Ratio | Sequence And Series | Mathematics 11

Geometric Sequence

A geometric sequence (also called a geometric progression, G.P.) is a sequence in which each term after the first is obtained by multiplying the preceding term by a fixed non-zero number called the common ratio $r$ .

Definition

A sequence $a_{1}, a_{2}, a_{3}, \dots, a_{n}$ is geometric if: $\frac{a _{n + 1}}{a _{n}} = r (constant for all n \geq 1)$

General Term

The $n$ -th term of a geometric sequence is: $a_{n} = a_{1} \cdot r^{n - 1}$ where $a_{1}$ is the first term and $r$ is the common ratio.

Example: Find the 6th term of the geometric sequence $2, 6, 18, 54, \dots$

$a_{1} = 2$ , $r = \frac{6}{2} = 3$
$a_{6} = 2 \cdot 3^{6 - 1} = 2 \cdot 243 = 486$

Finding the Common Ratio

For any geometric sequence, the common ratio is found by dividing any term by its preceding term: $r = \frac{a _{2}}{a _{1}} = \frac{a _{3}}{a _{2}} = \dots = \frac{a _{n}}{a _{n - 1}}$

Example: Find the common ratio of $5, 15, 45, 135, \dots$ $r = \frac{15}{5} = 3$

Harmonic Sequence

A harmonic sequence is a sequence whose reciprocals form an arithmetic sequence.

If $a_{1}, a_{2}, a_{3}, \dots$ is an arithmetic sequence with common difference $d$ , then: $\frac{1}{a _{1}}, \frac{1}{a _{2}}, \frac{1}{a _{3}}, \dots$ is a harmonic sequence.

General Term of a Harmonic Sequence

The $n$ -th term of a harmonic sequence is: $H_{n} = \frac{1}{a _{1} + ( n - 1 ) d}$ where $a_{1}$ is the first term of the corresponding arithmetic sequence and $d$ is the common difference.

Example: Find the 5th term of the harmonic sequence $1, \frac{1}{3}, \frac{1}{5}, \frac{1}{7}, \dots$

Corresponding A.P.: $1, 3, 5, 7, \dots$ with $a_{1} = 1$ , $d = 2$
$H_{5} = \frac{1}{1 + ( 5 - 1 ) ( 2 )} = \frac{1}{1 + 8} = \frac{1}{9}$

Harmonic Mean

The harmonic mean $H$ between two numbers $a$ and $b$ is: $H = \frac{2 ab}{a + b}$

This is the middle term of a harmonic sequence with $a$ and $b$ as the first and third terms.

Example: Find the harmonic mean between $3$ and $6$ . $H = \frac{2 ( 3 ) ( 6 )}{3 + 6} = \frac{36}{9} = 4$

Harmonic Series

The harmonic series is the sum of terms of a harmonic sequence: $\sum_{n = 1}^{N} \frac{1}{a _{1} + ( n - 1 ) d}$

Unlike geometric series, the harmonic series does not have a simple closed-form sum formula. Problems involving harmonic series are typically solved by converting to the corresponding arithmetic sequence.

Example: Find the sum of the first 4 terms of the harmonic sequence $\frac{1}{2}, \frac{1}{5}, \frac{1}{8}, \frac{1}{11}, \dots$

$S_{4} = \frac{1}{2} + \frac{1}{5} + \frac{1}{8} + \frac{1}{11}$ $= \frac{220 + 88 + 55 + 40}{440} = \frac{403}{440}$

Relationship Between A.M., G.M., and H.M.

For two positive numbers $a$ and $b$ :

Arithmetic Mean: $A = \frac{a + b}{2}$
Geometric Mean: $G = ab$
Harmonic Mean: $H = \frac{2 ab}{a + b}$

Important inequality: $A \geq G \geq H$

Also: $G^{2} = A \cdot H$ , i.e., the geometric mean is the geometric mean of the arithmetic and harmonic means.

Example: For $a = 4$ and $b = 9$ :

$A = \frac{4 + 9}{2} = 6.5$
$G = 36 = 6$
$H = \frac{2 ( 4 ) ( 9 )}{13} = \frac{72}{13} \approx 5.54$
Check: $G^{2} = 36 = A \cdot H = 6.5 \times \frac{72}{13} = 36$ ✓

Geometric Sequence

Definition

A sequence $a_{1}, a_{2}, a_{3}, \dots, a_{n}$ is geometric if: $\frac{a _{n + 1}}{a _{n}} = r (constant for all n \geq 1)$

General Term

The $n$ -th term of a geometric sequence is: $a_{n} = a_{1} \cdot r^{n - 1}$ where $a_{1}$ is the first term and $r$ is the common ratio.

Example: Find the 6th term of the geometric sequence $2, 6, 18, 54, \dots$

$a_{1} = 2$ , $r = \frac{6}{2} = 3$
$a_{6} = 2 \cdot 3^{6 - 1} = 2 \cdot 243 = 486$

Finding the Common Ratio

For any geometric sequence, the common ratio is found by dividing any term by its preceding term: $r = \frac{a _{2}}{a _{1}} = \frac{a _{3}}{a _{2}} = \dots = \frac{a _{n}}{a _{n - 1}}$

Example: Find the common ratio of $5, 15, 45, 135, \dots$ $r = \frac{15}{5} = 3$

Harmonic Sequence

A harmonic sequence is a sequence whose reciprocals form an arithmetic sequence.

If $a_{1}, a_{2}, a_{3}, \dots$ is an arithmetic sequence with common difference $d$ , then: $\frac{1}{a _{1}}, \frac{1}{a _{2}}, \frac{1}{a _{3}}, \dots$ is a harmonic sequence.

General Term of a Harmonic Sequence

The $n$ -th term of a harmonic sequence is: $H_{n} = \frac{1}{a _{1} + ( n - 1 ) d}$ where $a_{1}$ is the first term of the corresponding arithmetic sequence and $d$ is the common difference.

Example: Find the 5th term of the harmonic sequence $1, \frac{1}{3}, \frac{1}{5}, \frac{1}{7}, \dots$

Corresponding A.P.: $1, 3, 5, 7, \dots$ with $a_{1} = 1$ , $d = 2$
$H_{5} = \frac{1}{1 + ( 5 - 1 ) ( 2 )} = \frac{1}{1 + 8} = \frac{1}{9}$

Harmonic Mean

The harmonic mean $H$ between two numbers $a$ and $b$ is: $H = \frac{2 ab}{a + b}$

This is the middle term of a harmonic sequence with $a$ and $b$ as the first and third terms.

Example: Find the harmonic mean between $3$ and $6$ . $H = \frac{2 ( 3 ) ( 6 )}{3 + 6} = \frac{36}{9} = 4$

Harmonic Series

The harmonic series is the sum of terms of a harmonic sequence: $\sum_{n = 1}^{N} \frac{1}{a _{1} + ( n - 1 ) d}$

Example: Find the sum of the first 4 terms of the harmonic sequence $\frac{1}{2}, \frac{1}{5}, \frac{1}{8}, \frac{1}{11}, \dots$

$S_{4} = \frac{1}{2} + \frac{1}{5} + \frac{1}{8} + \frac{1}{11}$ $= \frac{220 + 88 + 55 + 40}{440} = \frac{403}{440}$

Relationship Between A.M., G.M., and H.M.

For two positive numbers $a$ and $b$ :

Arithmetic Mean: $A = \frac{a + b}{2}$
Geometric Mean: $G = ab$
Harmonic Mean: $H = \frac{2 ab}{a + b}$

Important inequality: $A \geq G \geq H$

Also: $G^{2} = A \cdot H$ , i.e., the geometric mean is the geometric mean of the arithmetic and harmonic means.

Example: For $a = 4$ and $b = 9$ :

$A = \frac{4 + 9}{2} = 6.5$
$G = 36 = 6$
$H = \frac{2 ( 4 ) ( 9 )}{13} = \frac{72}{13} \approx 5.54$
Check: $G^{2} = 36 = A \cdot H = 6.5 \times \frac{72}{13} = 36$ ✓