Jennifer just turned 23 and can save $500 per quarter, starting in three months. If Jennifer can earn 7% compounded quarterly, what age will she be when she accumulates $1,000,000?

This is a straight savings problem. The formula required is the one you use to find the sum of a geometric sequence. I find it useful to consider the last deposit first.

The last deposit earns no interest. It is $500.

The next-to-last deposit earns 1 quarter's interest, so it contributes

... $500 * (1 +.07/4) = 500*1.0175

to the sum.

The deposit before that contributes

... $500*1.0175²

to the sum.

Clearly, the sum is that of "n" terms of a geometric sequence with first term 500 and common ratio 1.0175. Your job is to find "n" that makes the total be $1 million and then convert that number of quarters to Jennifer's age.

The sum of "n" terms of a geometric sequence with first term "a" and common ratio "r" is given by

... S = a * (r^n - 1) / (r - 1)

We can solve this for n.

... S * (r - 1) / a + 1 = r^n

... n = log (S * (r - 1) / a) / log (r)

... n = log (10⁶ * (0.0175) / 500 + 1) / log (1.0175)

... n ≈ 206.559

The balance in Jennifer's account will reach $1,000,000 when Jennifer makes her 207th payment. She will be 74 3/4.