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?
Accepted Solution
A:
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*1.0175Β² 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(10βΆ*(0.0175)/500 +1) / log(1.0175) .. n β 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.