Chapter 11.2, Problem 17WE

To determine

To calculate:

The value of $t_{21}$ for following arithmetic sequence:

  $t_7=-19,t_{10}=-28$

Answer to Problem 17WE

The value of $t_{21}$ the arithmetic sequence is $-61$ .

Explanation of Solution

Given information:

  $t_7=-19,t_{10}=-28$

Formula used:

The $n^{th}$ term of an arithmetic sequence is given by the following expression:

  $t_n=t_1+\left(n-1\right)d$

Where,

First term: $t_1$

Common difference: $d$

Calculation:

The equations are formed as:

  $\begin{array}{l}{t}_7=t_1+\left(7-1\right)d\\ -19=t_1+6d\end{array}$

  $t_1+6d=-19$  ... (1)

And

  $\begin{array}{l}{t}_{10}=t_1+\left(10-1\right)d\\ -28=t_1+9d\end{array}$

  $t_1+9d=-28$  ... (2)

Now, $t_1$ is calculated by taking subtraction of both eqns.

  $\begin{array}{l}{t}_1+6d-t_1-9d=-19+28\\ -3d=9\\ d=-3\end{array}$

Now, put value of $d=-3$ in eq. (1):

  $\begin{array}{l}{t}_1+6d=-19\\ t_1+6\left(-3\right)=-19\\ t_1=-19+18\\ t_1=-1\end{array}$

Now, value of $t_{40}$ the arithmetic sequence is calculated as:

  $\begin{array}{l}{t}_{21}=t_1+\left(21-1\right)d\\ t_{21}=-1+20\left(-3\right)\\ t_{21}=-1-60\\ t_{21}=-61\end{array}$