Geometric Sequence | Math Help

A geometric sequence is also referred to as a geometric progression. Each term of a geometric sequence can be obtained from the previous one my multiplying by the same non-zero constant.

For example, $2, \ 6, \ 18, \ 54, \cdots$ is a geometric sequence as each term can be obtained by multiplying the previous term by $3$. Notice that $6 \div 2 = 18 \div 6 = 54 \div 18 = 3$, so each term divided by the previous one gives the same constant, this is often called a common ratio.

Algebraic Definition of a Geometric Sequence

$\dfrac{T_{n+1}}{T_{n}} = r$ for all positive integers $n$ where $r$ is a constant called the common ratio.

An Important Property of a Geometric Sequence

If $a, \ b$ and $c$ are any consecutive terms of a geometric sequence then $\dfrac{b}{a} = \dfrac{c}{b}$.
That is, $b^2 = ac$ and so $b = \pm \sqrt{ac}$ where $\sqrt{ac}$ is the geometric mean of $a$ and $c$.

The General Term Formula of a Geometric Sequence

Suppose the first term of a geometric sequence is $a$ and the common ratio is $r$.
\begin{aligned} \displaystyle T_1 &= a \\ T_2 &= T_1 \times r = ar \\ T_3 &= T_2 \times r = ar \times r = ar^2 \\ T_4 &= T_3 \times r = ar^2 \times r = ar^3 \\ &\text{and so on …} \\ T_n &= T_{n-1} \times r = ar^{n-2} \times r = ar^{n-1} \\ \therefore T_n &= ar^{n-1}\\ \end{aligned} \\

Practice Questions

Question 1

Show that the sequence $240,120,60,30, \cdots$ is geometric.

Question 2

Find the general term of a geometric sequence: $3, 6, 12, 24, \cdots$.

Question 3

Find $k$, if $k+1, 3k,$ and $5k+2$ consecutive terms of a geometric sequence.

Question 4

Find the general term of a geometric sequence its third term is 48 and its sixth term is -3072. 