Chapter 12, Problem 1T

To determine

The first six terms and the sixth partial term of the sequence.

Answer to Problem 1T

The value of first six terms is $1,6,15,28,45$ and 66. The value of partial sixth term is $161$.

Explanation of Solution

Given:

The $n^{th}$ term of the sequence is given as,

$a_n=2n^2-n$ (1)

Calculation:

Section 1:

The $n^{th}$ term of the sequence is given as,

$a_n=2n^2-n$.

Calculation:

To calculate the value of first six terms,

Solve the value of first term $a_1$,

Substitute 1 for n in equation (1),

$\begin{array}{c}{a}_1=2\cdot 1^2-1\\ =2-1\\ =1\end{array}$

The value of $a_1$ is 1.

Solve the value of second term $a_2$,

Substitute 2 for n in equation (1),

$\begin{array}{c}{a}_2=2\cdot 2^2-2\\ =2\cdot 4-2\\ =8-2\\ =6\end{array}$

The value of $a_2$ is 6.

Solve the value of third term $a_3$,

Substitute 3 for n in equation (1),

$\begin{array}{c}{a}_3=2\cdot 3^2-3\\ =18-3\\ =15\end{array}$

The value of $a_3$ is 15.

Solve the value of fourth term $a_4$,

Substitute 4 for n in equation (1),

$\begin{array}{c}{a}_4=2\cdot 4^2-4\\ =32-4\\ =28\end{array}$

The value of $a_4$ is 24.

Solve the value of fifth term $a_5$,

Substitute 5 for n in equation (1),

$\begin{array}{c}{a}_5=2\cdot 5^2-5\\ =50-5\\ =45\end{array}$

The value of $a_5$ is 45.

Solve the value of sixth term $a_6$,

Substitute 6 for n in equation (1),

$\begin{array}{c}{a}_6=2\cdot 6^2-6\\ =72-6\\ =66\end{array}$

The value of $a_6$ is 66.

Thus, the value of first six terms is $1,6,15,28,45$ and 66.

Section 2:

The n th partial sum of any sequence is denoted by $S_n$ is the sum of the first n terms. Thus, sixth partial sum of sequence is denoted by $S_6$ is the sum of first six terms.

Calculate the sixth partial sum of sequence by the formula,

$S_6=a_1+a_2+a_3+a_4+a_5+a_6$ (2)

Substitute 1 for $a_1$, 6 for $a_2$,15 for $a_3$, 28 for $a_4$ 45 for $a_5$ and 66 for $a_6$ in equation (3),

$\begin{array}{c}{S}_6=1+6+15+28+45+66\\ =161\end{array}$

Thus, the sixth partial sum $S_6$ is equal to 161.