Chapter 12, Problem 2T

To determine

The first six terms of the arithmetic sequence.

Answer to Problem 2T

The value of first six terms is $2,5,13,36,104$ and 307.

Explanation of Solution

Given:

The $a_1$ term is 2 and $n+1^{th}$ term of the sequence is given as,

$a_{n+1}=3a_n-n$ (1)

Calculation:

To calculate the value of first six terms,

Since the first term is given as $a_1=2$.

Solve the value of second term $a_2$,

Substitute 1 for n in equation (1),

$\begin{array}{c}{a}_{1+1}=3a_1-1\\ a_2=3\times 2-1\\ =6-1\\ =5\end{array}$

The value of $a_2$ is 5.

Solve the value of third term $a_3$,

Substitute 2 for n in equation (1),

$\begin{array}{c}{a}_{2+1}=3a_2-2\\ a_3=3\times 5-2\\ =15-2\\ =13\end{array}$

The value of $a_3$ is 13.

Solve the value of fourth term $a_4$,

Substitute 3 for n in equation (1),

$\begin{array}{c}{a}_{3+1}=3a_3-3\\ a_4=3\times 13-3\\ =39-3\\ =36\end{array}$

The value of $a_4$ is 36.

Solve the value of fifth term $a_5$,

Substitute 4 for n in equation (1),

$\begin{array}{c}{a}_{4+1}=3a_4-4\\ a_5=3\times 36-4\\ =108-4\\ =104\end{array}$

The value of $a_5$ is 104.

Solve the value of sixth term $a_6$,

Substitute 6 for n in equation (1),

$\begin{array}{c}{a}_{5+1}=3a_5-5\\ a_6=3\times 104-5\\ =312-5\\ =307\end{array}$

The value of $a_6$ is 307.

Thus, the value of first six terms is 2, 5, 13, 36 and 307.