Question

A computer software company decides to set aside $100,000 to develop
a new video game. It estimates that development will cost $955 the first week and
that expenses will increase by $65 each week.

a. After 25 weeks, how much of the development budget will be left?
b. How long can the company keep the development phase going before the budget
will not support another week of expenses?

Answer 1

a. First week is 955. Each subsequent week will also require a spend of at least 955, and an additional 65. So, total spend at the end of 25 weeks is 955*25 + 24*65 = $25,435. Which will leave $74,565.

b. 955*25 + 24*65 can be expressed as 955*25 + (25-1)*65... so that's 25*(955+65) - 65. Right? Right.

Why did we do that? To arrive at a general formula... looks like we can say that the amount of money spent in 'n' weeks is

n*1020 - 65

So what we're asking is how long before this hits 100000

so if n*1020 - 65 = 100000

then n = (100000+65)/1020 = 98.102 weeks

Which means we can't spend to 99 weeks. So it seems like 98 weeks is the max here.

We can verify this by doing what we did in answer "a". Total spend at the end of 98 weeks is 955*98 + 97*65 = 99,895

Seems close to $100K... what happens if we add another week? We'll have to spend another 955+65, which will take us to $100,915.

So there it is, the answer is 98 weeks, fer shure!