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:

5, 10, 20, 40, 80, 160, has a1 = 5, r = 2.

Here, we start with 5 and multiply by 2 each time to get the next number in the progression.

The n-th term of a GP is given by:

an = a1rn-1


Proof


Let's see how LiveMath can find the n-th term of a GP for us:

LIVEMath



Example:

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


Answer


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:

math expression

Here is how LiveMath can find a sum to n terms for us:

LIVEMath


Example:

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

chess
[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?


Here is the LiveMath solution to this problem.

LIVEMath


And now for the conventional answer:


Answer




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