Geometric Progressions

Geometric Progression

5.2

Geometric Progression

Definition

A Geometric Progression (GP) is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called common ratio, \(r\).

General Form

The \(n\)-th term of a geometric progression is given by:

\(T_n=ar^{(n-1)}\)

Where:
\(a=\) first term,
\(r=\) common ratio,
\(n=\) term number,
\(T_n=n\)-th term.

Common Ratio, \(r\)

Formula to find the common ratio, \(r\):

\(r=\dfrac{T_n}{T_{n-1}}\)

Example:
For the sequence \(2\), \(6\), \(18\), \(54\), the common ratio, \(r\):
\(\begin{aligned} r&=\dfrac{6}{2} \\ &=3. \end{aligned}\)

Sum of the First \(n\) Terms, \(S_n\)

The sum of the first \(n\) terms, \(S_n\), of a geometric progression is given by:

\(S_n=a\dfrac{1-r^n}{1-r}\) (if \(r\neq1\))

For \(r=1\), the sum is:

\(S_n=an\)

Sum to Infinity, \(S_\infin\)

The sum to infinity, \(S_\infin\), of a geometric progression, where \(|r|<1\), is given by:

\(S_\infin=\dfrac{a}{1-r}\)

Examples

Finding the \(n\)-th Term

Given \(a=3\), \(r=2\), find the \(5^{\text{th}}\) term.
Solution:
\(\begin{aligned} T_5&=3\times2^4 \\ &=3\times16 \\ &=48 .\end{aligned}\)

Finding the Sum of the First \(n\) Terms

Given \(a=5\), \(r=0.5\), find the sum of the first \(6\) terms.
Solution:
\(\begin{aligned} S_6&=5\dfrac{1-0.5^6}{1-0.5} \\ &=5\dfrac{1-0.015625}{0.5} \\ &=5\times1.96875\times2 \\ &=19.6875 .\end{aligned}\)