# 2. Geometric Progressions

by M. Bourne

A Geometric Progression (GP) is formed by multiplying a starting number (a1) 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:

an = a1rn−1

### Example 2

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

Continued below

## The Sum of a Geometric Progression

The sum to n terms of a GP means:

a1 + a1r + a1r2 + a1r3 + ... + a1rn-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"?

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