Question

You may use a spreadsheet for this one. If $60 is put in an account that gets 7% per year, and I add $15 at the end of each year, how much will I have at the end of 8 years?

Answer 1

I am assuming that you are depositing $60 right now, and that there are 9 terms in the series
Amount = 60(1.07)^8 + 75(1.07)^7 + 90(1.07)^6 + .... + 150(1.07)^2 + 165(1.07) + 180

It would take you only about 5 minutes on a good calculator with memory storage to actually calculate this, there are only 9 terms.
However, this is what is called an "algebraic-geometric" series.
Notice the coefficients form an arithmetic sequence, while the powers of 1.07 are a geometric series.

multiply each side by 1.07 and <b>line up the like powers</b>, we get
1.07(Amount)
= 60(1.07)^9 + 75(1.07)^8 + 90(1.07)^7 + ... + 165(1/07)^2 + 180(1.07)
Now subtract:
1.07Amount - Amount
= 60(1.07)^9 + 15(1.07)^8 + 15(1.07)^7 + 15(1.07)^6 +...+15(1.07) - 180
Amount(1.07 - 1) = 60(1.07)^9 + <b>[15(1.07)^8 + 15(1.07)^7 + 15(1.07)^6 +...+15(1.07)]</b> - 180

the highlighted part is a GS with a = 15(1.07), r = 1.07, and n = 8
value = 15(1.07)(1.07^8 - 1)/.07 = 164.67

Amount = (60(1.07)^9 + 164.67 - 180)/.07 = $1356.82

check my arithmetic, I should have written it out first.