# 2. Geometric Progressions

by M. Bourne

A Geometric Progression (GP) is formed by **multiplying** a
starting number (*a*_{1}) by a number *r*,
called the **common ratio.**

### Example 1

The progression `5, 10, 20, 40, 80, 160`, has first term `a_1= 5`, and common ratio `r = 2`.

In this example, we started with `5` and multiplied by `2` each time to get the next number in the progression.

## Formula for the `n`-th term of a GP

The *n*-th term of a geometric progression is given by:

a_{n}=a_{1}r^{n−1}

Explanation

The first term is

a_{1}

The second term is obtained by multiplying the first by *r*

a_{1}r

The third term is obtained by multiplying the second by *r*

a_{1}r^{2}

The fourth term is obtained by multiplying the third by *r*

a_{1}r^{3}

We continue this pattern and can see that in general, the *n*-th term is

a_{1}r^{n−1}

### Example 2

Find the 50^{th} term of the
geometric progression 5, 10, 20, 40, 80, ...

Answer

Since `a_1= 5`, `r = 2`, and using

a_{n}=a_{1}r^{n-1},

we have:

a_{50}= (5)(2^{50−1})= 2,814,749,767,106,560

≈ 2.81 × 10

^{15}

## The Sum of a Geometric Progression

The sum to *n* terms of a GP means:

a_{1}+a_{1}r+a_{1}r^{2}+a_{1}r^{3}+ ... +a_{1}r^{n-1}

We can show (using Proof by Induction) that this sum is equivalent to:

`S_n=(a_1(1-r^n))/(1-r)\ (r!=1)`

### Example 3

(We first saw this story in the Chapter Introduction).

[Image source.]

A king once promised a prince anything he wanted because he saved the princess's life. The prince requested one grain of rice on the first square of a chess board, `2` on the second, `4` on the third, `8` on the fourth square, etc.

How much rice is there if one grain of rice weighs `20\ "mg"`?

Answer

We need `1 + 2 + 4 + 8 + ... + 2^63`

Now `a_1= 1`, `r = 2`, `n = 64`.

Our formula for the sum to *n* terms says:

`S_n=(a_1(1-r^n))/(1-r)\ (r!=1)`

Substituting our values:

`S_64=(1(1-2^64))/(1-2)`

`=1.84467xx10^19\ "grains"`

Each grain weighs `20\ "mg" = 2 × 10^-5\ "kg" ` `= 2 × 10^-8\ "tonnes"`.

So the weight is

`(1.84467 × 10^19) × ` `(2 × 10^-8)\ "tonnes" ` `= 369\ "billion tonnes"`, so of course, the king cannot grant the Prince's wish.

**NOTE 1:** There are `1000\ "kg"` in one tonne.

**NOTE 2:** The world annual output of rice today is only `600` million (not **b**illion) tonnes!

**NOTE 3: **We are using the US/French 'billion' (`10^9`) and not the British 'billion' (`10^12`). [See Short and Long Scales.]