Chapter 9, Problem 12PS

To determine

Find the total income in 30 days.

Answer to Problem 12PS

  $\$10,737,418.23$

Explanation of Solution

Calculation:

Payment on first day is $\$0.01$

Payment on second day is $\$0.02$

Payment on third day is $\$0.04$

Thus the total income in $30$ days is

  $0.01+0.02+0.04+0.08+\mathrm{.....}$ (30^th day wages)

It can be written as

  $\begin{array}{l}0.02=2(0.01)\\ 0.04=2^2(0.01)\\ 008=2^3(0.01)\end{array}$

Thus 30^th day wage is $2^{29}(0.01)$

Total income is

  $0.01+2(0.01)+2^2\left(0.01\right)+\mathrm{...}+2^{29}\left(0.01\right)=0.01\left[1+2+2^2+2^3+\mathrm{...}+2^{29}\right]$

  $\begin{array}{l}=0.01\left[\frac{2^{30}-1}{2-1}\right]\left(S_n=\frac{a_1(1-r_n)}{1-r}in,G.P.\right)\\ =0.01\left(2^{30}-1\right)\\ =10,737,418.23\end{array}$

Hence, the total income in 30 days is $\$10,737,418.23$