Answer & Step-by-step explanation:
To find an expression for the account's value at the end of three years in terms of the growth factor, we need to consider the deposits made at the start of each year and the interest earned.
Let's break it down step by step:
1. Year 1: $500 is deposited at the start of the first year. At the end of the year, it will grow with the interest rate, which we'll denote as "r." So, the value at the end of the first year is $500 * (1 + r).
2. Year 2: In the second year, an additional $200 is deposited at the start of the year. The balance at the beginning of the second year (the end of the first year) is $500 * (1 + r). This balance will then grow with the interest rate for one year. So, at the end of the second year, the value is:
$500 * (1 + r) * (1 + r) + $200
We can simplify this to:
$500 * (1 + r)^2 + $200
3. Year 3: In the third year, an additional $600 is deposited at the start of the year. The balance at the beginning of the third year (the end of the second year) is $500 * (1 + r)^2 + $200. This balance will then grow with the interest rate for one year. So, at the end of the third year, the value is:
($500 * (1 + r)^2 + $200) * (1 + r) + $600
Now, we have an expression for the value of the account at the end of three years in terms of the growth factor "r":
V = ($500 * (1 + r)^2 + $200) * (1 + r) + $600
This expression represents the account's value at the end of three years with the given deposits and annual interest rate, and it's in terms of the growth factor "r."