Geometric Sequences
A geometric sequence is a sequence of numbers in which the ratio between each term an and the preceding term an-1 is constant. $$ q = \frac{a_n}{ a_{n-1} } $$ Here, n denotes a natural number.
This constant ratio is called the common ratio.
$$ q = \frac{a_n}{ a_{n-1} } $$
In practical terms, every term after the first is obtained by multiplying the previous term by q:
$$ a_n = a_{n-1} \cdot q $$
Alternatively, each term of the sequence can be found by dividing the next term by the common ratio.
$$ a_n = \frac{ a_{n+1}}{ q } $$
The value of q determines the behavior of the sequence.
- q>0
all terms have the same sign
- if 0<q<1
the sequence decreases when the terms are positive
(and increases when they are negative) - if q=1
the sequence is constant - if q>1
the sequence increases when the terms are positive
(and decreases when they are negative)
- if 0<q<1
- q<0
the terms alternate in sign
Note. A geometric sequence is defined only when all terms are nonzero, since the ratio involves division by an-1. Therefore, the common ratio must never be zero.
A geometric sequence may contain either finitely many or infinitely many terms. When the sequence is finite, we refer to the first and last elements as the first term and the last term.
Why is it called a geometric sequence? Each term is the geometric mean of its neighbors. $$ a_n = \sqrt{a_{n-1} \cdot a_{n+1}} $$
Example
Consider the sequence
$$ 2 \ , \ 4 \ , \ 8 \ , \ 16 \ , \ 32 \ , \ 64 \ \dots $$
This is a geometric sequence with infinitely many terms.
The common ratio is
$$ q = 2 $$
Note. Each term is obtained by multiplying the previous one by q $$ a_n = a_{n-1} \cdot q $$ $$ 4 = 2 \cdot 2 \\ 8 = 4 \cdot 2 \\ 16 = 8 \cdot 2 \\ 32 = 16 \cdot 2 \\ 64 = 32 \cdot 2 \\ $$
Since q>0 and all terms are positive, the sequence is increasing.
Example 2
Consider the sequence
$$ 64 \ , \ 32 \ , \ 16 \ , \ 8 \ , \ 4 \ , \ 2 \ \dots $$
This is a finite geometric sequence.
The first term is 64 and the last term is 2.
The common ratio is
$$ q = \frac{1}{2} $$
Since 0<q<1 and the terms are positive, the sequence is decreasing.
Note. The terms are obtained as follows $$ a_n = a_{n-1} \cdot q $$ $$ 32 = 64 \cdot \frac{1}{2} \\ 16 = 32 \cdot \frac{1}{2} \\ 8 = 16 \cdot \frac{1}{2} \\ 4 = 8 \cdot \frac{1}{2} \\ 2 = 4 \cdot \frac{1}{2} \\ $$
Example 3
Consider the sequence
$$ 5 \ , \ 5 \ , \ 5 \ , \ 5 \ , \ 5 \ , \ 5 \ , \ \dots $$
This is a constant geometric sequence.
In this case, the common ratio is
$$ q = 1 $$
Example 4
Consider the sequence
$$ 2 \ , \ -4 \ , \ 8 \ , \ -16 \ , \ 32 \ , \ -64 \ , \ \dots $$
This sequence alternates in sign because the common ratio is negative.
$$ q = -2 $$
The sequence is neither increasing nor decreasing.
Note. The terms are obtained as follows $$ a_n = a_{n-1} \cdot q $$ $$ -4 = 2 \cdot (-2) \\ 8 = (-4) \cdot (-2) \\ -16 = 8 \cdot (-2) \\ 32 = (-16) \cdot (-2) \\ -64 = 32 \cdot (-2) \\ $$
Key properties
Here are some useful results that make geometric sequences easy to work with:
- The general term is given by $$ a_n = a_1 \cdot q^{n-1} $$ for n ≥ 1
Example. Let q=3 and a1=2 $$ 2 \ , \ 6 \ , \ 18 \ , \ 54 \ , \ 162 \ , \ 486 \ \dots $$ The fourth term is $$ a_4 = 2 \cdot 3^{3} = 54 $$ Why does this work? Each step multiplies by q, so after n-1 steps we obtain $$ a_n = a_1 \cdot q^{n-1} $$
- Any two terms are related by $$ a_x = a_y \cdot q^{x-y} $$
Example. In the sequence $$ 2 \ , \ 6 \ , \ 18 \ , \ 54 \ , \ 162 \ , \ 486 \ \dots $$ $$ a_2 = a_4 \cdot q^{2-4} $$ $$ 6 = 54 \cdot 3^{-2} = 54 \cdot \frac{1}{9} = 6 $$
- Each term is the geometric mean of its neighbors $$ a_n = \sqrt{a_{n-1} \cdot a_{n+1}} $$
Example. $$ a_3 = \sqrt{6 \cdot 54} = 18 $$
- In the first n terms, the product of two terms equidistant from the ends is constant and equal to a1⋅an
Example. $$ 2 \ , \ 6 \ , \ 18 \ , \ 54 \ , \ 162 $$ $$ a_1 \cdot a_5 = 324 $$ $$ a_2 \cdot a_4 = 324 $$
- The product of the first n terms is given by $$ P_n = \sqrt{(a_1 \cdot a_n)^n} $$
Example. $$ 2 \ , \ 6 \ , \ 18 \ , \ 54 $$ $$ P_4 = 11664 $$ $$ P_4 = \sqrt{(2 \cdot 54)^4} = 11664 $$
- Sum of the first n terms
The sum of the first $ n $ terms of a geometric sequence with common ratio $ q \ne 1 $ is given by the formula $$ S_n = a_1 \cdot \frac{q^n - 1}{q - 1} $$ where $ a_1 $ denotes the first term of the sequence. The same formula can also be written as $$ S_n = a_1 \cdot \frac{1 - q^n}{1 - q} $$ The two expressions are equivalent, since one can be obtained from the other by multiplying both the numerator and the denominator by $ -1 $.Example. Consider a geometric sequence with first term $ a_1 = 2 $ and common ratio $ q = 3 $. The first four terms ($ n = 4 $) are: $$ 2,\ 6,\ 18,\ 54 $$ Compute the sum directly: $$ S_4 = 2 + 6 + 18 + 54 = 80 $$ Now apply the formula: $$ S_n = a_1 \cdot \frac{q^n - 1}{q - 1} $$ $$ S_4 = 2 \cdot \frac{3^4 - 1}{3 - 1} = 2 \cdot \frac{81 - 1}{2} = 80 $$ The result matches the direct computation.
And so on.
