A bank pays \( 6 \% \) interest, compounded semiannually. Use the appropriate formula to find how much should be deposited (in \( \$ \) ) now to yield an annuity payment of \( \$ 300 \) at the beginning of each six months, |for 18 years. (Round your answer to the nearest cent.)

We are given a bank that pays an annual interest rate of \(6\%\), compounded semiannually. This means that the interest rate per six‐month period is \[ i=\frac{6\%}{2}=3\%=0.03. \] Since the term is 18 years and there are 2 periods per year, the total number of periods is \[ n=18\times2=36. \] Because the annuity payment of \(\$300\) is made at the beginning of each period, we are dealing with an annuity due. The present value \(PV\) of an annuity due is given by the formula \[ PV=P\cdot\frac{1-(1+i)^{-n}}{i}\cdot(1+i), \] where \(P=\$300\) is the payment per period. Substituting the numbers into the formula, we have \[ \begin{aligned} PV &=300 \cdot \frac{1-(1.03)^{-36}}{0.03}\cdot1.03. \end{aligned} \] Let's break this down step by step. 1. Compute \((1.03)^{-36}\). Writing it as \[ (1.03)^{-36}=\frac{1}{(1.03)^{36}}, \] and noting that \[ (1.03)^{36}\approx2.8971, \] we have \[ (1.03)^{-36}\approx\frac{1}{2.8971}\approx0.3450. \] 2. Next, compute the annuity immediate factor: \[ \frac{1-(1.03)^{-36}}{0.03} \approx \frac{1-0.3450}{0.03} = \frac{0.6550}{0.03}\approx21.8333. \] 3. Multiply by the payment and by the extra factor for an annuity due: \[ PV \approx300\times21.8333\times1.03. \] First, \[ 300\times21.8333\approx6549.99, \] then \[ 6549.99\times1.03\approx6744.50. \] Thus, the amount that should be deposited now is approximately \(\$6,744.50\).