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.
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
Find the 50th term of the geometric progression 5, 10, 20, 40, 80, ...
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:
(We first saw this story in the Chapter Introduction).
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"`?