20.4. Infinite Geometric Progressions
20.4.1. Sum of an Infinite Geometric Progression
Before we derive a formula for the sum of an infinite geometric progression, let us explore the idea graphically and numerically.
We have already determined that the sum of n terms of any geometric progression with a first term a and a common ratio r is
Thus a graph of sn versus n should tell us about how the sum changes as n increases.
Example 33:Graph the sum sn versus n for the GP
for n = 0 to 20. Solution: Here, a = 1 and r = 1.2, so
The sum sn is graphed as shown. Note that the sum continues to increase. We say that this progression diverges. |
Screen for Example 33. Tick marks are 5 units apart.
20.4.2. Decreasing GP
If the common ratio r in a GP is less than 1, each term in the progression will be less than those preceding it. Such progression is called a decreasing GP.
Example 34:Graph the sum sn versus n for the GP
for n = 0 to 20. Solution: Here, a = 1 and r = 0.8, so
The sum is graphed as shown. Note that the sum appears to reach a ... |
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