Chapter 11.2, Problem 15WE

To determine

To calculate:

The value of $t_1$ for following arithmetic sequence:

  $t_5=24,t_9=40$

Answer to Problem 15WE

The value of $t_1$ the arithmetic sequence is $8$ .

Explanation of Solution

Given information:

  $t_5=24,t_9=40$

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:

  $\begin{array}{l}{t}_5=t_1+\left(5-1\right)d\\ 24=t_1+4d\end{array}$

  $t_1+4d=24$  ... (1)

And

  $\begin{array}{l}{t}_9=t_1+\left(9-1\right)d\\ 40=t_1+8d\end{array}$

  $t_1+8d=40$  ... (2)

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

  $\begin{array}{l}{t}_1+4d-t_1-8d=24-40\\ -4d=-16\\ d=4\end{array}$

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

  $\begin{array}{l}{t}_1+4d=24\\ t_1+4\left(4\right)=24\\ t_1=24-16\\ t_1=8\end{array}$