Question

Find a recurrence relation for the amount of money in a savings account after n years if the interest rate is 6 percent and $50 is added at the start of each year.

Answer 1

`a_n = a_(n-1) (1.06) + 50`

Explanation:

Suppose,
`a_0` is initial amount in the saving account,

Here, the annual interest rate is 6% and additional amount in each year is $ 50,

So, the amount after one year,

`a_1 = a_0 + 6\%\text{ of }a_0 + 50 = a_0 + 0.06a_0 + 50 = a_0(1.06) + 50`

Amount after 2 years,

`a_2 = a_1 + 6\%\text{ of }a_1 + 50 = a_1(1.06) + 50`

Amount after 3 years,

`a_3 = a_2 + 6\%\text{ of }a_2 + 50 = a_2(1.06) + 50`

................................., so on....

Hence, by following the pattern,

The amount after n years,

`a_n = a_(n-1) (1.06) + 50`

Which is the required recurrence relation for the amount of money in a savings account