The geometric sequence

Sequences, grade 12 (4 units) Lesson 12 of 20Learning

The geometric sequence

A constant ratio

A geometric sequence is a sequence in which each term equals the term before it times a fixed number, $\displaystyle q$ – the common ratio:

\displaystyle a_{n+1}=a_n\cdot q

$\displaystyle 3, 6, 12, 24, \ldots$ – $\displaystyle {q}={2}$ ;
$\displaystyle 80, 40, 20, 10, \ldots$ – $\displaystyle {q}={\frac{1}{2}}$ ;
$\displaystyle 2, -6, 18, -54, \ldots$ – $\displaystyle {q}={-3}$ .

$\displaystyle q$ is found from any two neighbouring terms: $\displaystyle {q}={\frac{a_{n+1}}{a_n}}$ . In a geometric sequence $\displaystyle a_1\neq0$ and $\displaystyle q\neq0$ , so no term is 0.

Four ways to show a geometric sequence

On its first day a video had 5 views, and each day it had twice as many views as the day before.

In a table:

$\displaystyle n$	1	2	3	4	5
$\displaystyle a_n$	5	10	20	40	80

By a recursion rule: $\displaystyle {a_1}={5}$ , $\displaystyle {a_{n+1}}={2a_n}$ .
On a graph – the points $\displaystyle (n, a_n)$ :

\displaystyle (1,5)

\displaystyle (2,10)

\displaystyle (3,20)

\displaystyle (4,40)

\displaystyle (5,80)

\displaystyle y=5\cdot2^{x-1}

The points are not on a line: with each step to the right the term doubles, so each rise is bigger than the one before. The graph of a geometric sequence is points on the graph of an exponential function (the dashed one) – a discrete graph.

The general term

From $\displaystyle a_1$ to $\displaystyle a_n$ we multiply by $\displaystyle q$ , $\displaystyle n-1$ times: $\displaystyle {a_2}={a_1q}$ , $\displaystyle {a_3}={a_1q^2}$ , $\displaystyle {a_4}={a_1q^3}$ , and so for every place:

\displaystyle a_n

\displaystyle =

\displaystyle a_1\cdot q^{n-1}

In the example: on day ten $\displaystyle {a_{10}}={5\cdot2^9}={2560}$ views.

Increasing, decreasing, alternating signs

The sign and the size of $\displaystyle q$ decide how the sequence behaves:

$\displaystyle q<0$ : the signs alternate – positive, negative, positive… – and the sequence neither increases nor decreases: $\displaystyle 2, -6, 18, -54, \ldots$
$\displaystyle q>0$ : all the terms have the sign of $\displaystyle a_1$ .
$\displaystyle a_1>0$ : with $\displaystyle q>1$ the sequence increases ( $\displaystyle 3, 6, 12, \ldots$ ), and with $\displaystyle 0<q<1$ it decreases ( $\displaystyle 80, 40, 20, \ldots$ ).
$\displaystyle a_1<0$ : the other way round – with $\displaystyle q>1$ it decreases ( $\displaystyle -3, -6, -12, \ldots$ ), and with $\displaystyle 0<q<1$ it increases ( $\displaystyle -80, -40, -20, \ldots$ ).
$\displaystyle {q}={1}$ : the sequence is constant.

Exercise 1

Which of the sequences are geometric? For a geometric one, write $\displaystyle q$ .

$\displaystyle 5, 15, 45, 135, \ldots$
$\displaystyle 64, 16, 4, 1, \ldots$
$\displaystyle 1, 4, 9, 16, \ldots$
$\displaystyle 3, -6, 12, -24, \ldots$

Exercise 2

The geometric sequence: $\displaystyle 5, 10, 20, \ldots$
Calculate $\displaystyle a_8$ .

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 3

The geometric sequence: $\displaystyle 486, 162, 54, \ldots$
Calculate $\displaystyle a_7$ .

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 4

The geometric sequence: $\displaystyle 2, -6, 18, \ldots$

Calculate $\displaystyle a_6$ .

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Is $\displaystyle a_{21}$ positive or negative? Explain.

Formulas: sequences, grade 12 (4 units)

Open as a page

A sequence and the sum of its first terms

The term from the sum – $\displaystyle S_n$ the sum of the first $\displaystyle n$ terms
$\displaystyle S_n-S_{n-1}$ $\displaystyle =$ $\displaystyle a_n$ $\displaystyle n\geq2$
The first term is the sum of one term
$\displaystyle a_1$ $\displaystyle =$ $\displaystyle S_1$

The arithmetic sequence

The recursion rule – $\displaystyle d$ the common difference
$\displaystyle a_{n+1}=a_n+d$
The general term
$\displaystyle a_n$ $\displaystyle =$ $\displaystyle a_1+(n-1)d$
The arithmetic mean of the neighbours
$\displaystyle a_n$ $\displaystyle =$ $\displaystyle \frac{a_{n-1}+a_{n+1}}{2}$ $\displaystyle n\geq2$
The sum of the first $\displaystyle n$ terms
$\displaystyle S_n$ $\displaystyle =$ $\displaystyle \frac{n(a_1+a_n)}{2}$
The sum without the last term
$\displaystyle S_n$ $\displaystyle =$ $\displaystyle \frac{n(2a_1+(n-1)d)}{2}$

The geometric sequence

The recursion rule – $\displaystyle q$ the common ratio
$\displaystyle a_{n+1}=a_n\cdot q$
The general term
$\displaystyle a_n$ $\displaystyle =$ $\displaystyle a_1\cdot q^{n-1}$
The geometric mean: a term’s square is the product of its neighbours
$\displaystyle a_n^2$ $\displaystyle =$ $\displaystyle a_{n-1}\cdot a_{n+1}$ $\displaystyle n\geq2$
The sum of the first $\displaystyle n$ terms
$\displaystyle S_n$ $\displaystyle =$ $\displaystyle \frac{a_1(q^n-1)}{q-1}$ $\displaystyle q\neq1$

The infinite geometric sequence

The sum of the infinite sequence
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{a_1}{1-q}$ $\displaystyle -1<q<1,\ q\neq0$