SOLUTION: Explain how the common ratio for a geometric sequence with positive terms determines whether the terms increase or decrease.

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Question 283866: Explain how the common ratio for a geometric sequence with positive terms determines whether the terms increase or decrease.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
A geometric sequence is a sequence of the form

a, ar, ar%5E2, ar%5E3, ....


where 'a' is the starting term and 'r' is the common ratio. If 'a' is positive, then the value of 'r' will determine whether the terms will increase or decrease. If 'r' is greater than 1 (ie r%3E1), then the terms will increase, while on the other hand, if 'r' is less than 1 (r%3C1), then the terms will decrease. Finally, if 'r' is equal to 1, then the terms will remain the same.


If r%3E1, then we can write 'r' as r=1%2Bk where k%3E1. Plug this into the sequence above to get:

a, a%281%2Bk%29, a%281%2Bk%29%5E2, a%281%2Bk%29%5E3, ....


Now expand and distribute


a, a%2Bak, a%2B2ak%2Bak%5E2, a%2B3ak%5E2%2B3ak%2Bak%5E3, ....


Notice how the second term a%2Bak is bigger than 'a' since 'ak' is positive ('a' is positive and so is 'k'). Symbolically, this can be proven by showing that a%3Ca%2Bak simplifies to ak%3E0 (which is true, as shown above). Similar arguments can be made to show that a%2B2ak%2Bak%5E2 is bigger than a%2Bak. Similarly, we can show that a%2B3ak%5E2%2B3ak%2Bak%5E3 is greater than a%2B2ak%2Bak%5E2, and so on.


For r%3C1, we can write 'r' as r=1-k where 0%3Ck%3C1 (we want to keep 'r' positive). I'll let you plug in 1-k for 'r'. Use similar arguments shown above to show that each term will get smaller.


Note: if r=1, then the sequence

a, ar, ar%5E2, ar%5E3, ....

becomes


a, a%281%29, a%281%29%5E2, a%281%29%5E3, ....



which simplifies to


a, a, a, a, ....