Chapter 12, Problem 2CT

To determine

To find: The first five terms of each sequence.

Answer to Problem 2CT

The first five terms of the given sequence are $4,14,44,134$ and $404$ .

Explanation of Solution

Given information:

The given terms are $a_1=4,a_n=3a_n+2$ .

Calculation:

Substitute $2$ for $n$ in $a_n=3a_{n-1}+2$ and simplify.

  $a_2=3a_1+2$

Replace $a_1$ with $4$ and evaluate.

  $\begin{array}{c}{a}_2=3\left(4\right)+2\\ =12+2\\ =14\end{array}$

Thus, the second term is $14$ .

Similarly, substitute $3,4,$ and $5$ to find the third, fourth, and fifth terms.

  $\begin{array}{c}{a}_3=3a_2+2\\ =3\left(14\right)+2\\ =42+2\\ =44\end{array}$

  $\begin{array}{c}{a}_4=3a_3+2\\ =3\left(44\right)+2\\ =132+2\\ =134\end{array}$

  $\begin{array}{c}a5=3a_4+2\\ =3\left(134\right)+2\\ =402+2\\ =404\end{array}$

Therefore, the first five terms of the given sequence are $4,14,44,134$ and $404$ .

Therefore, the first five terms of the sequence are $0,\frac{3}{10},\frac{8}{11},\frac{5}{4}$ and $\frac{24}{13}$ .