91Ó°ÊÓ

Find an expression for \(\bar{a}_{\vec{n} |}\) if \(\delta_{i}=\frac{1}{1+t}\)

There is \(\$ 40,000\) in a fund which is accumulating at \(4 \%\) per annum convertible continuously. If money is withdrawn continuously at the rate of \(\$ 2400\) per annum, how long will the fund last?

$$\text { Simplify } \sum_{t=1}^{20}(t+5) v^{t}$$.

The following payments are made under an annuity: 10 at the end of the fifth year, 9 at the end of the sixth year, decreasing by 1 each year until nothing is paid. Show that the present value is $$\frac{10-a_{\overline{14}}+a_{4]}(1-10 i)}{i}$$

A perpetuity-immediate has annual payments of \(1,3,5,7 \ldots\) If the present value of the sixth and seventh payments are equal, find the present value of the perpetuity.

An annuity-immediate has semiannual payments of \(800,750,700, \ldots, 350,\) at \(f^{(2)}=.16 .\) If \(a_{101.08}=A,\) find the present value of the annuity in terms of \(A\).

Annual deposits are made into a fund at the beginning of each year for 10 years. The first 5 deposits are \(\$ 1000\) each and deposits increase by \(5 \%\) per year therafter. If the fund earns \(8 \%\) effective, find the accumulated value at the end of 10 years. Answer to the nearest dollar.

Find the present value of a 20 -year annuity with annual payments which pays \(\$ 600\) immediately and each subsequent payment is \(5 \%\) greater than the preceding payment. The annual effective rate of interest is \(10.25 \%\). Answer to the nearest dollar.

a) Find the sum of the payments in \((I a)_{2}^{(12)}\) b) Find the sum of the payments in \(\left(I^{(12)} a\right)_{2}^{(12)}\)

Show that the present value of a perpetuity on which payments are 1 at the end of the 5 th and 6 th years, 2 at the end of the 7 th and 8 th years, 3 at the end of the 9 th and 10 th years is $$\frac{v^{4}}{i-v d}$$