Chapter 11.2, Problem 18WE

To determine

To calculate:

The value of $t_3$ for following arithmetic sequence:

  $t_{10}=41,t_{15}=61$

Answer to Problem 18WE

The value of $t_3$ the arithmetic sequence is $13$ .

Explanation of Solution

Given information:

  $t_{10}=41,t_{15}=61$

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:

  $\begin{array}{l}{t}_{10}=t_1+\left(10-1\right)d\\ 41=t_1+9d\end{array}$

  $t_1+9d=41$  ... (1)

And

  $\begin{array}{l}{t}_{15}=t_1+\left(15-1\right)d\\ 61=t_1+14d\end{array}$

  $t_1+14d=61$  ... (2)

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

  $\begin{array}{l}{t}_1+9d-t_1-14d=41-61\\ -5d=-20\\ d=4\end{array}$

Now, put value of $d=4$ in eq. (1):

  $\begin{array}{l}{t}_1+9d=41\\ t_1+9\left(4\right)=41\\ t_1=41-36\\ t_1=5\end{array}$

Now, value of $t_3$ the arithmetic sequence is calculated as:

  $\begin{array}{l}{t}_3=t_1+\left(3-1\right)d\\ t_3=5+4\left(2\right)\\ t_3=5+8\\ t_3=13\end{array}$