Meg opened a savings account that pays compound interest. This table shows the value of her account at different times after her initial deposit.
Time (years) Value
1 $2,550
2 $2,601
3 $2,653.02
Which function describes the account's value, in dollars, after t years?
The function that describes the account's value after t years can be found using the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = final amount (the account's value after t years)
P = principal amount (the initial deposit)
r = annual interest rate (as a decimal)
n = number of times interest is compounded per year
t = time in years
We can use the given values in the table to find the values of P, r, and n:
From year 1 to year 2, the account's value increases by $51. This means that the interest earned in one year is $51. So, r/n = 51/P.
From year 2 to year 3, the account's value increases by $52.02. This means that the interest earned in one year is $52.02 - $51 = $1.02. So, r/n = 1.02/P.
Simplifying the two equations:
51/P = r/n ...(1)
1.02/P = r/n ...(2)
Multiplying equation (1) by 1.02:
51(1.02)/P = (r/n)(1.02)
52.02/P = 1.02(r/n) ...(3)
Equating equations (2) and (3):
1.02/P = 52.02/P
1.02 = 52.02
This is not possible, which means there is an error in the given data. The interest rate or compounding frequency cannot be calculated with the given values.
Therefore, a function describing the account's value cannot be determined with the given information.