Over the course of 10 years, someone puts $400 per month into an investment account that earns 6% per year (compounds monthly). What is the value of the account at the end of 10 years?

Let's work this out month by month for a few months, find the pattern, then solve.

We're putting $400 per month into an account that is earning 6% per year with the interest compounding monthly.

Let's work out the interest rate first: we have `(6%)/"year"`, which gives us `(6%)/(12 "months")` or `1/2%` per month.

We start month 1 by adding $400 to our account. We earn interest over the month, so have `400 xx (1+.005) = 402`.

In month 2, we add $400 to the account, `402 + 400=802` and at the end of the month we have another round of interest: `(802) xx (1+.005) = 806.01`

Is there a pattern yet?

`806.01=802(1+.005)=(400+402)(1+.005)=(400+400(1+.005))(1+.005)`

Let's do one more and I think it will be clear:

In month 3, we add $400 to the account, `806.01 + 400=1206.01` and at the end of the month we have another round of interest: `(1206.01) xx (1+.005) = 1212.04`

Let's look at how we got here:

`1212.04=(1206.01)(1+.005)=(400+806.01)(1+.005)=(400+(400+400(1+.005))(1+.005))(1+.005)`

This is `sum_(n=1)^120((400)(1+.005))^n`

and far easier to compute on a spreadsheet or a financial calculator.

The answer I came up with on my spreadsheet is $65, 879. 50. I also worked it out on an online financial calculator and it returned $65, 551. 74 - which is what my sheet returned at the start of Month 120 but I calculated it through with that last month's interest (and so I think my answer is correct).

Here is where that financial calculator that I mentioned is located:

http://www.financialcalculator.org/investing/interest-calculator