Chapter 9, Problem 1SE

To determine

The fifth sequence from the sequence $s_n=3s_{n-1}+n^2$ for $n\ge 1$ recurrence relation and the given initial condition.

Answer to Problem 1SE

The fifth sequence $\underset{\_}{s_5=829}$.

Explanation of Solution

The given Sequence is $s_n=3s_{n-1}+n^2$ for $n\ge 1$.

Obtain the fifth sequence as follows.

Compute the first sequence as follows.

$\begin{array}{c}{s}_1=3s_{1-1}+{\left(1\right)}^2\\ s_1=3s_0+1\\ =3\left(2\right)+1\\ =7\end{array}$

Compute the second sequence as follows.

$\begin{array}{c}{s}_2=3s_{2-1}+{\left(2\right)}^2\\ s_2=3s_1+4\\ =3\left(7\right)+4\\ =25\end{array}$

Compute the third sequence as follows.

$\begin{array}{c}{s}_3=3s_{3-1}+{\left(3\right)}^2\\ s_3=3s_2+9\\ =3\left(25\right)+9\\ =84\end{array}$

Compute the fourth sequence as follows.

$\begin{array}{c}{s}_4=3s_{4-1}+{\left(4\right)}^2\\ s_4=3s_3+16\\ =3\left(84\right)+16\\ =268\end{array}$

Compute the fifth sequence as follows.

$\begin{array}{c}{s}_5=3s_{5-1}+{\left(5\right)}^2\\ s_5=3s_4+25\\ =3\left(268\right)+25\\ =829\end{array}$

Thus, the fifth sequence $\underset{\_}{s_5=829}$.