Ben deposits $5000 now into an account that earns 7.5 percent interest compounded annually. He then deposits $1000 per year at the end of the first and second years.

How much will the account contain 10 years after the initial deposit?

Multiply $5000 by 10 years

10,000

To calculate the amount in the account after 10 years, we need to calculate the future value of the initial deposit as well as the annual deposits made during the first two years.

Step 1: Calculate the future value of the initial deposit after 10 years.
Using the compound interest formula:
Future Value = Initial Deposit * (1 + Interest Rate) ^ Number of Years
Future Value = $5000 * (1 + 0.075) ^ 10
Future Value = $5000 * (1.075) ^ 10
Future Value = $5000 * 1.71872640567
Future Value = $8593.63

Step 2: Calculate the future value of the annual deposits made during the first two years.
Annual Deposit = $1000

For the first deposit made at the end of the first year:
Future Value_1 = Annual Deposit * (1 + Interest Rate) ^ Number of Years
Future Value_1 = $1000 * (1 + 0.075) ^ 9
Future Value_1 = $1000 * (1.075) ^ 9
Future Value_1 = $1000 * 1.68748482728
Future Value_1 = $1687.48

For the second deposit made at the end of the second year:
Future Value_2 = Annual Deposit * (1 + Interest Rate) ^ Number of Years
Future Value_2 = $1000 * (1 + 0.075) ^ 8
Future Value_2 = $1000 * (1.075) ^ 8
Future Value_2 = $1000 * 1.5668682935
Future Value_2 = $1566.87

Step 3: Calculate the total future value after 10 years.
Total Future Value = Future Value (Initial Deposit) + Future Value_1 (First Annual Deposit) + Future Value_2 (Second Annual Deposit)
Total Future Value = $8593.63 + $1687.48 + $1566.87
Total Future Value = $11847.98

Therefore, the account will contain $11,847.98 after 10 years.

To calculate the future value of Ben's account after 10 years, we can use the formula for compound interest, which is:

FV = PV * (1 + r)^n

Where:
FV = Future Value
PV = Present Value (initial deposit)
r = Interest rate per period (compounded annually)
n = Number of periods

Let's break down the problem step by step:

Step 1: Calculate the future value of the initial deposit.
Ben initially deposits $5000. We can substitute these values into the formula:

FV1 = 5000 * (1 + 0.075)^10

Step 2: Calculate the future value of the additional deposits.
Ben deposits $1000 per year at the end of the first and second years, for a total of 8 additional deposits. Each of these deposits will accumulate interest for a different number of years. We can sum up the future values of these deposits using the compound interest formula:

FV2 = 1000 * [(1 + 0.075)^9 + (1 + 0.075)^8 + ... + (1 + 0.075)^1]

Step 3: Calculate the total future value of the account after 10 years.
To find the total future value, we need to add both the initial deposit and the additional deposits:

Total future value = FV1 + FV2

Finally, let's substitute the values into the formulas and calculate the result:

FV1 = $5000 * (1 + 0.075)^10 = $5000 * 1.61051 = $8052.57 (rounded to two decimal places)

Now, let's calculate FV2. We'll use the formula for the sum of a geometric series:

FV2 = $1000 * [(1.075^9 - 1) / (1.075 - 1)] = $1000 * [1.91346 / 0.075] = $25,512.73 (rounded to two decimal places)

Finally, let's find the total future value:

Total future value = $8052.57 + $25,512.73 = $33,565.30

Therefore, after 10 years, the account will contain approximately $33,565.30.