Chapter 7.8, Problem 58STP

To determine

To write explicit and a recursive formulas for a geometric sequence $27,36,48,64,\mathrm{.......}$ .

Answer to Problem 58STP

The explicit rule is $a_n=48{\left(\frac{3}{4}\right)}^{n-1}$ and the recursive rule is $a_n=\left(\frac{3}{4}\right)a_{n-1},a_1=48,\text{where}n=2,3,4,5,\mathrm{.......}$

Explanation of Solution

Given information :

Givengeometric sequence $27,36,48,64,\mathrm{.......}$

Asa geometric sequence $27,36,48,64,\mathrm{.......}$ .

Where $a_1=27$ and common ratio $r=\frac{a_2}{a_1}=\frac{36}{27}=\frac{4}{3}$

Thus, the explicit rule or $n^{th}\text{term}$ for a geometric sequence is $a_n=a_1{r}^{n-1}$

  $\therefore a_n=27{\left(\frac{4}{3}\right)}^{n-1}$

Thus, recursive rule for geometric sequence is $a_n=r{a}_{n-1}$

  $\Rightarrow a_n=\left(\frac{4}{3}\right)a_{n-1},a_1=27,\text{where}n=2,3,4,5,\mathrm{.......}$

Hence, the explicit rule is $a_n=27{\left(\frac{4}{3}\right)}^{n-1}$ and the recursive rule is $\Rightarrow a_n=\left(\frac{4}{3}\right)a_{n-1},a_1=27,\text{where}n=2,3,4,5,\mathrm{.......}$