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
Let's see how LiveMath can find the n-th term of a GP for us:
Example:
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:
Here is how LiveMath can find a sum to n terms for us:
Example:
(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?
Here is the LiveMath solution to this problem.
And now for the conventional answer:
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