SOLUTION: Joanie wants to save $90 for a gift. She begins by saving $.25 the first week. Each week she saves three times as much as the week before. In how many weeks will she have saved at

Question 34488: Joanie wants to save $90 for a gift. She begins by saving $.25 the first week. Each week she saves three times as much as the week before. In how many weeks will she have saved at least $90?
Answer by mukhopadhyay(490) (Show Source): You can put this solution on YOUR website!
This forms a geometric sequence (.25, .75, 2.25,....) with initial savings as a quarter and a ratio of 3.
Sum of the savings must be greater than or equals $90.
Let the number of weeks required to save $90 is n
Sum = Initial[(Ratio)^n - 1]/(Ratio - 1)
= .25(3^n - 1)/(3-1) = .25/2 (3^n - 1);
We need to find minimum n for which .25/2 (3^n - 1) > = 90
=> (3^n - 1) >= 180/.25
=> (3^n - 1) >= 720
=> 3^n >= 721
3^5 = 243; 3^6 = 729;
Answer: 6 weeks
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