a$.$ Draw time lines for $(1)$ a $$100$ lump sum cash flow at the end of Year $2,(2)$ an ordinary annuity of $$100$ per year for $3$ years, and $(3)$ an uneven cash flow stream of $-$$$50,$$$100,$ $$75,$ and $$50$ at the end of Years $0$ through $3.$

b$.(1)$ What's the future value of an initial $$100$ after $3$ years if it is invested in an account paying $10\%$ annual interest?

$(2)$ What's the present value of $$100$ to be received in $3$ years if the appropriate interest rate is $10\%?$

c$.$ We sometimes need to find out how long it will take a sum of money $($or something else, such as earnings, population, or prices$)$ to grow to some specified amount. For example, if a company's sales are growing at a rate of $20\%$ per year, how long will it take sales to double?

d$.$ If you want an investment to double in $3$ years, what interest rate must it earn?

e$.$ What's the difference between an ordinary annuity and an annuity due? What type of annuity is shown below? How would you change the time line to show the other type of annuity?