Question

On your fifteenth birthday you discover you have a rich aunt sally aunt sally is a very generous women and wants to provide for your future she has decided that she will initially give you $1 then $2 the next year and so on doubling the amount each year until your 30. Write an exponential function to show the amount of money you will receive

Answer 1

Write out the information in the question

$\begin{gathered} \text{Let the first amount you where give=a} \\ \end{gathered}$$\text{first year=\ 1, second year=\ 2}$

The progression continued in the form

$1, $2, $4, $8...

hence this is a geometric progression

since r=2

The amount will be receive 15 years i.e 30-15

The amount of money that will be received until you are 30 is the sum of the progression

$\begin{gathered} s_n=\frac{a(r^n-1)}{r-1} \\ \text{where r=2, a=1} \end{gathered}$$\begin{gathered} S_n=\frac{1(2^n-1)}{2-1} \\ =2^n-1 \\ \text{where n=number of years} \end{gathered}$

Hence the exponential function is given by

$2^n-1$