Chapter ISG, Problem 7.10.9P

To determine

To write: An explicit and recursive formula for the given situation.

Answer to Problem 7.10.9P

It will take $24$ weeks to prepare for a $26.2$ miles race, recursive formula is $a_n=(1.10)a_{n-1}$ , explicit formula is $a_n=3\times {(1.10)}^{n-1}$ .

Explanation of Solution

Given information: Situation is regarding marathon preparation.

Calculation:

From the given information, sequence is,

  $3,3.30,3.63,\mathrm{3.993.......}$

Recursive Formula:

Each succeeding term is $110\%$ of the previous term,

  $\Rightarrow a_n=(1.10)a_{n-1}$

Explicit formula:

To calculate $n^{th}$ term of a GP formula is $T_n=a{r}^{n-1}$ , for this situation,

  $\Rightarrow a_n=3\times {(1.10)}^{n-1}$

When $a_n=26.2$ ,

  $\begin{array}{l}$ 3\times {(1.10)}^{n-1}=26.2$$ {(1.10)}^{n-1}=(\frac{26.2}{3})$$ \log {(1.10)}^{n-1}=\log (8.73)(Taking\log onbothsides)$\\ (n-1)\log (1.10)=\log (8.73)\\ \\ (n-1)(0.04139)=0.94118\\ \\ n-1=22.738\\ \\ n=23.738\end{array}$

As $n$ should be a natural number and its value is greater than $23$ , so it will take $24$ weeks to prepare for a $26.2$ miles race.

Thus, it will take $24$ weeks to prepare for a $26.2$ miles race, recursive formula is $a_n=(1.10)a_{n-1}$ , explicit formula is $a_n=3\times {(1.10)}^{n-1}$ .