Chapter 7.10, Problem 1BGP

To determine

To write: the explicit formula for the given recursive formula.

Answer to Problem 1BGP

The explicit formula is $a_n=-4{(1.5)}^{n-1}$ .

Explanation of Solution

Given information:

Consider $a_1=-4,a_n=\frac{3}{2}{a}_{n-1},n\ge 2$

Calculation:

Expand the sequence as

  $\begin{array}{l}{a}_1=-4,\\ a_2=\frac{3}{2}{a}_1=-6\\ a_3=\frac{3}{2}{a}_2=-9\\ a_4=\frac{3}{2}{a}_3=-13.5\\ .\\ .\end{array}$

The sequence is $-4,-6,-9,-13.5,\mathrm{......}$

Common ratio $r=-6/-4=1.5$

So, the sequence is in geometric progression.

Then find the general term.

  $\begin{array}{l}{a}_n=a_1{r}^{n-1}\\ $ =-4{(1.5)}^{n-1}$\end{array}$

Hence, the explicit formula is $a_n=-4{(1.5)}^{n-1}$ .