Question

the present value of an annuity-immediate with annual payments of 1 for n years is 40; and the present value of an annuity-immediate with annual payments of 1 for 3n years is 70. calculate the accumulated value of an annuity-immediate with annual payments of 1 for 2n years.

Answer 1

To find the accumulated value, calculate the interest rate i using the given present value of annuities formula for n and 3n years. Once i is found, use the annuity formula to compute the accumulated value for 2n years.

Step-by-step explanation:

To solve the problem, we need to find the interest rate from the first two annuities and then use it to calculate the accumulated value of an annuity-immediate for 2n years. We start by setting up the present value formulas for annuities-immediate:

PV1 = (1 -
`(1 + i)^(-n)`) / i = 40
PV2 = (1 -
`(1 + i)^(-3n)`) / i = 70
We have two equations here with two unknowns, n and i. Once we find the interest rate i, we can calculate the accumulated value for 2n years using the formula:

A = R * (
`(1 + i)^(2n)` - 1) / i

Where A is the accumulated value and R is the annual payment which is given as 1 in this case.

As the question requires us to calculate the accumulated value and not present or future value, we will find the interest rate i from the given present values and then apply it to the above formula to get the accumulated value for 2n years.