Explanation:

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 $400 \times \left(1 + .005\right) = 402$.

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: $\left(802\right) \times \left(1 + .005\right) = 806.01$Is there a pattern yet? $806.01 = 802 \left(1 + .005\right) = \left(400 + 402\right) \left(1 + .005\right) = \left(400 + 400 \left(1 + .005\right)\right) \left(1 + .005\right)$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: $\left(1206.01\right) \times \left(1 + .005\right) = 1212.04$

Let's look at how we got here:

$1212.04 = \left(1206.01\right) \left(1 + .005\right) = \left(400 + 806.01\right) \left(1 + .005\right) = \left(400 + \left(400 + 400 \left(1 + .005\right)\right) \left(1 + .005\right)\right) \left(1 + .005\right)$

This is ${\sum}_{n = 1}^{120} {\left(\left(400\right) \left(1 + .005\right)\right)}^{n}$

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