1.2.3 Geometric Sequences & Series

Geometric Sequences

In a geometric sequence, there is a common ratio, r, between consecutive terms in the sequence
- For example, 2, 6, 18, 54, 162, … is a sequence with the rule ‘start at two and multiply each number by three’
  - The first term, u₁, is 2
  - The common ratio, r, is 3
A geometric sequence can be increasing (r > 1) or decreasing (0 < r < 1)
If the common ratio is a negative number the terms will alternate between positive and negative values
- For example, 1, -4, 16, -64, 256, … is a sequence with the rule ‘start at one and multiply each number by negative four’
  - - The first term, u₁, is 1
    - The common ratio, r, is -4
Each term of a geometric sequence is referred to by the letter u with a subscript determining its place in the sequence

$u_{n} = u_{1} r^{n - 1}$

- Where $u_{1}$ is the first term, and $r$ is the common ratio
- This formula allows you to find any term in the geometric sequence
- It is given in the formula booklet, you do not need to know how to derive it
Enter the information you have into the formula and use your GDC to find the value of the term
Sometimes you will be given a term and asked to find the first term or the common ratio
- Substitute the information into the formula and solve the equation
  - You could use your GDC for this
Sometimes you will be given two or more consecutive terms and asked to find both the first term and the common ratio
- Find the common ratio by dividing a term by the one before it
- Substitute this and one of the terms into the formula to find the first term
Sometimes you may be given a term and the formula for the n^th term and asked to find the value of n
- You can solve these using logarithms on your GDC

You will sometimes need to use logarithms to answer geometric sequences questions
- Make sure you are confident doing this
- Practice using your GDC for different types of questions

The sixth term, $u_{6}$ , of a geometric sequence is 486 and the seventh term, $u_{7}$ , is 1458.

Find,

the common ratio,

r

, of the sequence,

ai-sl-1-2-3-geo-seq-i

ii)

the first term of the sequence,

u_{1}

ai-sl-1-2-3-geo-seq-ii

A geometric series is the sum of a certain number of terms in a geometric sequence
- For the geometric sequence 2, 6, 18, 54, … the geometric series is 2 + 6 + 18 + 54 + …
The following formulae will let you find the sum of the first n terms of a geometric series:

$S_{n} = \frac{u_{1} (r^{n} - 1)}{r - 1} = \frac{u_{1} (1 - r^{n})}{1 - r}$

- - $u_{1}$ is the first term
  - $r$ is the common ratio
- Both formulae are given in the formula booklet, you do not need to know how to derive them
You can use whichever formula is more convenient for a given question
- The first version of the formula is more convenient if $r > 1$ and the second is more convenient if $r < 1$
A question will often give you the sum of a certain number of terms and ask you to find the value of the first term, the common ratio, or the number of terms within the sequence
- Substitute the information into the formula and solve the equation
  - You could use your GDC for this

The geometric series formulae are in the formula booklet, you don't need to memorise them
- Make sure you can locate them quickly in the formula booklet

A geometric sequence has $u_{1} = 25$ and $r = 0.8$ . Find the value of $u_{5}$ and $S_{5}$ .

ai-sl-1-2-3-geo-series