How do I solve this Senior Finite problem on annuities?

Question

Raul Vasquez, a 25-year-old professional, puts $750 in a retirement fund at the end of each quarter until he reaches age 60. The account pays 8% interest compounded annually. (a) How much will be in the account when he is 60? (b) If Raul makes no further deposits after age 60, how much will he have for retirement at the age of 65?

Answer 1

This has two parts, theA interest on the initial deposit ($750) and the interest on the recurring deposits.

 

The amount on the first deposit:

 

Since in the first year, Raul puts in $2,250

 

A = p(1 + r/n)^nt , where A = amount at the end, r is the rate, n is number times deposit\calculated, t = time in years.

A = 2250 (1 + 0.08)⁴⁰

A = 2250 (1.08)⁴⁰

A = 2250 · 21.72

A = 48870

 

A = d{((1 + r)^t - 1) /r}, where d = the monthly deposit. Here t = 34 since the first three deposits were included above.

 

A = 750{((1 + 0.08)³⁴ -1) / 0.08}

A = 750 {(13.6901 - 1) / 0.08}

A = 750 (12.6901 / 0.08)

A= 750 · 158.62

A = 118970

 

The total amount is $167,840.