SOLUTION: Please help me solve this word problem: You are saving $590 a week to buy a new computer. Your brother is saving at a weekly rate of 4^{x-2}, if x = the number of weeks of saving

Algebra ->  Coordinate Systems and Linear Equations  -> Linear Equations and Systems Word Problems -> SOLUTION: Please help me solve this word problem: You are saving $590 a week to buy a new computer. Your brother is saving at a weekly rate of 4^{x-2}, if x = the number of weeks of saving      Log On


   

Question 1097962: Please help me solve this word problem:
You are saving $590 a week to buy a new computer. Your brother is saving at a weekly rate of 4^{x-2}, if x = the number of weeks of saving. What is the first week that your brother will have more money saved than you?
When I solved it I got 7 but then realized that each week is increasing by 590 for the person. I don't know what to do.
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
The "You" person is saving %22%24590%22 per week,
so after n weeks, the saving for that person (in $) would be
590n .
The savings grow linearly.

The amounts the brother would save (in $) would be
4%5E%281-2%29=4%5E%28-1%29=0.25 for week 1,
4%5E%282-2%29=4%5E0=1 for week 2,
4%5E%283-2%29=4%5E1=4 for week 3,
4%5E%284-2%29=4%5E2=16 for week 4,
and so on.
The weekly savings form a geometric sequence
(or geometric progression, or GP, as you wish to call it).
The dirst term b=0.25 ,
the common ratio is r=4 ,
and the sum of the first n terms is
.
The brother's savings grow exponentially.

At the beginning the "you" person will have more saved,
but the brother is a faster saver,
and would soon have more saved than the you person.

That would happen when
4%5E%28n-1%29%2F3-0.25%2F3%3E590n .
It is easy to get to the answer by calculating both sides for a few values of n .

CALCULATING FOR A FEW n VALUES:
For n=6 , 590n=590%2A6=3540 and %284%5E%28n-1%29-0.25%29%2F3=%284%5E5-0.25%29%2F3=%281024-0.25%29%2F3=1023.75%2F3=341.25 ,
so the brother does not have the most savings.
For n=8 , 590n=590%2A8=4720 and %284%5E%28n-1%29-0.25%29%2F3=%284%5E7-0.25%29%2F3=%2896384-0.25%29%2F3=96383.75%2F3=5461.25 ,
and the brother have more savings than the "you" person.
What about for n=7 ?
For n=7 , 590n=590%2A7=5310 and %284%5E%28n-1%29-0.25%29%2F3=%284%5E6-0.25%29%2F3=%284096-0.25%29%2F3=4095.75%2F3=1365.25 , and the "you" person still has nore savings than the brother.
So the first time that the brother has more money saved than the you person is
when the brother has highlight%288%29 weeks worth of savings.