Chapter 3.6, Problem 27PPS

To determine

To determine the first term of an arithmetic sequence.

Answer to Problem 27PPS

  $77$

Explanation of Solution

Given:

The twelfth term of an arithmetic sequence is $(t_{12})22$ .

The common difference of the arithmetic sequence is $(d)-5$ .

Formula used:

nth term of any arithmetic progression is given by the formula:

  $t_n=a+(n-1)d$

  $t_n=$ nth term

  $a=$ The first term of the arithmetic progression

  $d=$ The common difference of the arithmetic progression

Calculation / Explanation:

Let the first term of the arithmetic sequence is a .

Using the above formula,

  $\Rightarrow a_n=a+(n-1)d$

  $\Rightarrow a_n=a+(n-1)(-5)$  $[\because d=-5]$

  $\Rightarrow a_{12}=a+(12-1)(-5)$

  $\Rightarrow 22=a+11(-5)$ [The fourth term of an arithmetic sequence is $(t_{12})22$ .]

  $\begin{array}{l}\Rightarrow 22=a-55\\ \Rightarrow a=22+55\\ \Rightarrow a=77\end{array}$

Conclusion:

Therefore, the first term of the arithmetic sequence is $77$ .