Geometric Sequences

Geometric Sequence Definition

Geometric Sequences are sequences where each term is obtained by multiplying the preceding term by a certain constant factor, which is often called $\textit{common ratio}$. A geometric sequence is also referred to as a $\textit{geometric progression}$.
Geometric Sequences

Geometric Sequences Formula

David expects $10$% increase per month to deposit to his account. A $10$% increase per month would mean that the amount would increase by a constant factor of $(1+0.1)$ or $1.1$.
He starts off with $100 initially.
By the beginning of the second month he will expect $$100 \times 1.1 = $$$110$, by the start of the third month he would expect $$110 \times 1.1 = $$$100 \times 1.1^2 = $$$121$, and so on.
This is an example of a geometric sequence

The first term is $10$, and the common factor is $1.1$, which represents a $10$% increase on the previous term.
\( \begin{align} \displaystyle u_{n+1} &= u_{n} \times 1.1 \\ u_{1} &= \$100 \\ u_{2} &= \$100 \times 1.1 \\ u_{3} &= \$100 \times 1.1^2 \\ u_{4} &= \$100 \times 1.1^3 \\ &\cdots \\ u_{n} &= \$100 \times 1.1^{n-1} \\ \end{align} \)

For a geometric sequence: $$\require{color} \color{red}u_{n} = u_{1}r^{n-1}$$ where $\require{color} \color{red}u_{1}$ is the first term and $\color{red}r$ the common ratio, given by $$\require{color} \color{red}\dfrac{u_{n+1}}{u_{n}} = r$$

Geometric Mean

If we consider three consecutive terms in a geometric sequence $\{x,y,z\}$ then $$\dfrac{y}{x} = \dfrac{z}{y} = r$$ where $r$ is the common ratio.
Thus the middle term, $y$, called the $\textit{geometric mean}$, can be calculated in terms of the outer two terms, $x$ and $z$. $$y^2 = xz$$ A geometric sequence is a sequence of numbers for which the ratio of successive terms is the same. $$\require{color} \color{red} \dfrac{u_{2}}{u_{1}} = \dfrac{u_{3}}{u_{2}} = \dfrac{u_{4}}{u_{3}} = \cdots = r$$

Example 1

Determine whether the sequence $\{2,10, 50, 250,\cdots\}$ is geometric.

Example 2

Determine whether the sequence $\{4,-8,16,-32,64,\cdots\}$ is geometric.

Example 3

Find the $n$th term and the $12$th in the geometric sequence where the first term is $3$ and the fourth term is $-24$.

Example 4

The first three terms of a geometric sequence are $2, 6$ and $18$. Which numbered term would be the first to exceed $1\ 000\ 000$ in this sequence?