Chapter 5.3, Problem 14P

To determine

To Find: How many Dollars would you receive on ${31}^{st}$ day.

Answer to Problem 14P

Amount of wage received on ${31}^{th}$ day $=1000000000.0$ dollars

Explanation of Solution

Given information:

Receiving wages of

  $1$ cent on First day

  $2$ cents on Second day

  $4$ cents of Third day

And so on, with each day’s wages twice the previous days.

Receiving wages of

  $1$ cent on First day

  $2$ cents on Second day

  $4$ cents of Third day

And so on, with each day’s wages twice the previous days.

This is following a Geometric sequence

  $1,2,4,\mathrm{....}$

First term of sequence $=a=1$

As given each day’s wages twice the previous days

Therefore,

Common ratio $=r=2$

We need to find Amount of wage received on ${31}^{st}$ day.

It means

  $n=31$

We know

  $n^{th}$ term of Geometric series is

  $a_n=a{r}^{n-1}$

On substituting values

We get

  $\begin{array}{l}{a}_n=a{r}^{n-1}\\ \Rightarrow a_{31}=\left(1\right)2^{31-1}\\ \Rightarrow a_{31}=2^{30}\end{array}$

Therefore,

Amount of wage received on ${31}^{th}$ day $=2^{30}$ dollars

Therefore,

On using approximation $2^{10}\approx {10}^3$

We get

  $\Rightarrow$ Amount of wage received on ${31}^{th}$ day $={\left(2^{10}\right)}^3$

  $\Rightarrow$ Amount of wage received on ${31}^{th}$ day $={\left({10}^3\right)}^3$

  $\Rightarrow$ Amount of wage received on ${31}^{th}$ day $={10}^{3\times 3}$

  $\Rightarrow$ Amount of wage received on ${31}^{th}$ day $={10}^9$

  $\Rightarrow$ Amount of wage received on ${31}^{th}$ day $=1000000000.0$

Therefore,

Amount of wage received on ${31}^{th}$ day $=1000000000.0$ dollars