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# 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

Explanation

The first term is

a1

The second term is obtained by multiplying the first by r

a1r

The third term is obtained by multiplying the second by r

a1r2

The fourth term is obtained by multiplying the third by r

a1r3

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

a1rn−1

### Example 2

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

Since a_1= 5, r = 2, and using

an = a1rn-1,

we have:

a50 = (5)(250−1)

= 2,814,749,767,106,560

≈ 2.81 × 1015

## 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"?

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 billion) tonnes!

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

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