Chapter 6.6, Problem 48E

To determine

To find $2^{nd},5^{th}and{10}^{th}$ terms of the sequence $f(1)=-1,f(n)=6f(n-1)$ .

Answer to Problem 48E

The $2^{nd},5^{th}\text{and}{10}^{th}$ terms of given sequence are $-6,-6^4=-1296\text{and}-6^9=-10077696$ respectively.

Explanation of Solution

Given information:

Given function $f$ that represents a sequence is $f(1)=-1,f(n)=6f(n-1)$

The given sequence $f(1)=-1,f(n)=6f(n-1)$ is recurrence rule of an geometric sequence with first term $a_1=-1$ and common ratio $r=6$

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

  $\Rightarrow a_n=\left(-1\right)6^{n-1}$

For $n=2$ , the second term is $a_2=\left(-1\right)6^{2-1}=-6$ .

For $n=5$ , the second term is $a_5=\left(-1\right)6^{5-1}=-6^4=-1296$ .

For $n=10$ , the second term is $a_5=\left(-1\right)6^{10-1}=-6^9=-10077696$ .

Hence, the $2^{nd},5^{th}and{10}^{th}$ terms of given sequence are $-6,-6^4=-1296\text{and}-6^9=-10077696$ respectively.