Question

Alex wants to start an IRA that will have $1,250,675 in it when he retires in 30 years. How much should he invest semiannually in his IRA to do this if the interest is 4.5% compounded semiannually? Assume an Annuity Due. Round to the nearest cent.

Answer 1

https://www.financeformulas.net/Future-Value-of-Annuity-Due.html

The formula we'll use is the future value of annuity due
The formula is
\[FV = (1+r)*P*\left[ \frac{(1+r)^n-1}{r}\right]\]
Keep in mind that this r value is not the annual rate, and instead it is the semiannual rate (since the deposit periods are semiannual)

An annual rate of 4.5% cuts in half to 2.25%
so r = 0.0225

n = number of payment periods (ie number of deposits)
n = 30*2 = 60 payments will be deposited
P = unknown payment per period, which is what we want to solve for
FV = 1,250,675 = amount we want in the account after 30 years

So,
\[FV = (1+r)*P*\left[ \frac{(1+r)^n-1}{r}\right]\]
\[1250675 = (1+0.025)*P*\left[ \frac{(1+0.025)^{60}-1}{0.025}\right]\]
I'll let you solve for P