What is the size of payments that must be deposited at the beginning of each 6-month period in an account that pays 5.6%?

Question

What is the size of payments that must be deposited at the beginning of each 6-month period in an account that pays 5.6%? compounded semiannually, so that the account will have a future value of $190,000 at the end of 16 years?

Answer 1

The answer is $3,644.95 based on the video solution.

Answer 2

You have 32 payments compounded semi-annually. The first payment is compounded 32 times at half of the rate of 5.6%, next one 31 times, and so on down the last payment compounded once.

The future value equals the payment (P) times the annuity. It looks like this:

To see what the calculation looks like paste this code into https://latexeditor.lagrida.com/

190,000 &= P\biggl[\biggl(1+\frac{.056}{2}\biggr)^{32}+\cdots+\biggl(1+\frac{.056}{2}\biggr)^{2}+\biggl(1+\frac{.056}{2}\biggr)\biggr]\\

190,000 &= P\bigg(\frac{1.028^{33}-1.028}{.028}\biggr)\\

P&=3644.95