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

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) (Show Source): You can put this solution on YOUR website!
The "You" person is saving per week,
so after weeks, the saving for that person (in $) would be
.
The savings grow linearly.

The amounts the brother would save (in $) would be
for week 1,
for week 2,
for week 3,
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 ,
the common ratio is ,
and the sum of the first 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
.
It is easy to get to the answer by calculating both sides for a few values of .

CALCULATING FOR A FEW VALUES:
For , and ,
so the brother does not have the most savings.
For , and ,
and the brother have more savings than the "you" person.
What about for ?
For , and , 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 weeks worth of savings.
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