Question

16–17. {Use of Tech} Periodic savingsSuppose you deposit m dollars at the beginning of every month in a savings account that earns a monthly interest rate of r, which is the annual interest rate divided by 12 (for example, if the annual interest rate is 2.4%, r = 0.024/12 = 0.002). For an initial investment of m dollars, the amount of money in your account at the beginning of the second month is the sum of your second deposit and your initial deposit plus interest, or m + m(1 + r). Continuing in this fashion, it can be shown that the amount of money in your account after n months isAₙ = m + m(1 + r) + ⋯ + m(1 + r)ⁿ⁻¹.Use geometric sums to determine the amount of money in your savings account after 5 years (60 months) using the given monthly deposit and interest rate.17. Monthly deposits of $250 at a monthly interest rate of 0.2%

Accepted Answer

Identify the variables given in the problem: the monthly deposit amount $$m = 250$$, the monthly interest rate $$r = 0.002 ($$which is 0.2%), and the total number of months $$n = 60 (5 $$years). Recognize that the total amount in the account after $$n$$ months is given by the sum of a geometric series:  
$$A_n = m + m(1 + r) + m(1 + r)^2 + \cdots + m(1 + r)^{n-1}.$$ Factor out $$m$$ from the sum to write it as:  
$$A_n = m \left[1 + (1 + r) + (1 + r)^2 + \cdots + (1 + r)^{n-1}\right].$$ Use the formula for the sum of a geometric series with first term $$a = 1$$ and common ratio $$q = 1 + r$$:  
$$\sum_{k=0}^{n-1} q^k = \frac{q^n - 1}{q - 1}.$$  
Substitute $$q = 1 + r to $$get:  
$$A_n = m \cdot \frac{(1 + r)^n - 1}{r}.$$ Substitute the known values $$m = 250$$, $$r = 0.002$$, and $$n = 60$$ into the formula to express the amount after 5 years:  
$$A_{60} = 250 \cdot \frac{(1 + 0.002)^{60} - 1}{0.002}.$$  
This expression can now be evaluated to find the total amount in the account.

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Key Concepts

Geometric Series and Sums

Compound Interest with Periodic Deposits

Converting Annual Interest Rate to Monthly Rate