We need to find the size of the payments that must be deposited at the beginning of each 6-month period in an account that pays 5.2% compounded semiannually, so that the account will have a future value of $130,000 at the end of 18 years. This is an annuity due problem.

1. Problem Description

We need to find the size of the payments that must be deposited at the beginning of each 6-month period in an account that pays 5.2% compounded semiannually, so that the account will have a future value of $130,000 at the end of 18 years. This is an annuity due problem.

2. Solution Steps

The formula for the future value of an annuity due is:

$FV = PMT \times \frac{(1+i)^n - 1}{i} \times (1+i)$

Where:

$FV$ = Future Value = $130,000

$PMT$ = Payment amount (what we want to find)

$i$ = interest rate per period

$n$ = number of periods

The annual interest rate is 5.2%, and it is compounded semiannually, so the interest rate per period is:

$i = \frac{5.2\%}{2} = \frac{0.052}{2} = 0.026$

The number of years is 18, and payments are made semiannually, so the number of periods is:

$n = 18 \times 2 = 36$

Now we can plug the values into the formula:

$130,000 = PMT \times \frac{(1+0.026)^{36} - 1}{0.026} \times (1+0.026)$

$130,000 = PMT \times \frac{(1.026)^{36} - 1}{0.026} \times (1.026)$

$130,000 = PMT \times \frac{2.4455 - 1}{0.026} \times 1.026$

$130,000 = PMT \times \frac{1.4455}{0.026} \times 1.026$

$130,000 = PMT \times 55.5962 \times 1.026$

$130,000 = PMT \times 56.993$

$PMT = \frac{130,000}{56.993} = 2280.98$

3. Final Answer

$2280.98