$2\mathrm{.}34$ Determine the present worth of a geometric gradient series with a cash flow of $$50,000$ in year $1$ and increases of $6\%$ each year through year $8.$ The interest rate is $10\%$ per year.

$2\mathrm{.}34P_g=50,000\frac{1-[\frac{1+0\mathrm{.}06}{1+0\mathrm{.}10}{]}^8}{0\mathrm{.}10-0\mathrm{.}06}$

$=$$$320,573$

$2\mathrm{.}35$ Determine the difference in the present worth values of the following two commodity contracts at an interest rate of $8\%$ per year. Contract $1$ has a cost of $$10,000$ in year $1$; costs will escalate at a rate of $4\%$ per year for $10$ years. Contract $2$ has the same cost in year $1,$ but costs will escalate at $6\%$ per year for $11$ years.

$2\mathrm{.}35P_{g1}=10,000\frac{1-[\frac{1+0\mathrm{.}04}{1+0\mathrm{.}08}{]}^{10}}{0\mathrm{.}08-0\mathrm{.}04}$

$=$$$78,590$

$P_{g2}=10,000\frac{1-[\frac{1+0\mathrm{.}06}{1+0\mathrm{.}08}{]}^{11}}{0\mathrm{.}08-0\mathrm{.}06}$

$=$$$92,926$

Difference $=$$$14,336$

$3\mathrm{.}3$ The net cash flow associated with development and sale of a new product is shown. Determine the present worth at an interest rate of $12\%$ per year. The cash flow is in $$1000$ units. Show $($ a $)$ hand and $($ b $)$ spreadsheet solutions.

$]=[-120(\frac{P}{F},12\%,1)-100(\frac{P}{F},12\%,2)-40(\frac{P}{F},12\%,3)+50(\frac{P}{A},12\%,2)(\frac{P}{F},12\%,3)$

$($b$)$ Enter cash flows in$,$ say, column A$,$ and use the function $=NPV(12\%,A2$:$A10)**1000$ to display $$-71,308.$

$3\mathrm{.}4$ Standby power for water utility pumps and other electrical devices is provided by diesel$-$powered generators. As an alternative, the utility can use natural gas to power the generators, but it will be a few years before the gas is available at remote sites. The utility estimates that by switching to gas, it will save $$22,000$ per year, starting $3$ years from now. At an interest rate of $8\%$ per year, determine the present worth in year $0$ of the projected savings that will occur in years $3$ through $10.$

$]=[22,000(\frac{P}{A},8\%,8)(\frac{P}{F},8\%,2)$