Question

An individual deposits an annual bonus into a savings account that pays 6% interest compounded annually. The size of the bonus increases by $5,000 each year, and the initial bonus amount was $20,000. Determine how much will be in the account immediately after the fifth deposit.

Answer 1

Answer:

Immediately after the fifth deposit the individual will have $54,950 in his account.

Explanation:

For each year you have to calculate the total savings that the indivual has in the account.

The first year, denoted by $Y_{0}$, the individual deposits $20,000 in his account. At the end of the year the interests are accrued on that principal, and the individual also deposits $5,000 more that will bear interests next year. So we have:

$Y_{0}= 20,000$

$Y_{1}= 20,000*(1+0.06)+ 5,000 = 26,200$

And for each year we calculate the total savings accumulated, using the savings of the previous year as this period's principal:

$Y_{2}= 26,200*(1+0.06)+ 5,000 = 32,772$

$Y_{3}= 32,772*(1+0.06)+ 5,000 = 39,738.32$

$Y_{4}= 39,738.32*(1+0.06)+ 5,000 = 47,122.62$

$Y_{5}= 47,122.62*(1+0.06)+ 5,000 = 54,949.98$

Therefore the answer is $54,949.98.

In general the formula used for each period is the following:

$P_{n} = P_{n-1}*(1+r)+D$

Where:

$P_{n}$ are the total savings for the current period,

$P_{n-1}$ are the total savings from last period,

$r$ is the interest rate,

$D$ are the monthly deposits made into the savings account.

We further know that $P_{0}= 20,000$.