Chapter 12.1, Problem 12CFU

To determine

To calculate: The common difference for the arithmetic sequence for which first term is $100$ , and twelfth term is $34$ .

Answer to Problem 12CFU

The common difference for the arithmetic sequence is $d=-6$ .

Explanation of Solution

Given information:

The arithmetic sequence has first term as $a_1=100$ , and twelfth term as $a_{12}=34$ .

Formula used:

A sequence with first term as $a_1$ and next term written as sum of preceding term and common difference then it is called an arithmetic sequence. It is denoted as $a_1,a_1+d,a_1+2d+\dots$ .

The $n\text{th}$ term of the arithmetic sequence is $a_n=a_1+\left(n-1\right)d$ .

Calculation:

Consider the arithmetic sequence that has first term as $a_1=100$ , and twelfth term as $a_{12}=34$ .

Recall that a sequence with first term as $a_1$ and next term written as sum of preceding term and common difference then it is called an arithmetic sequence. It is denoted as $a_1,a_1+d,a_1+2d+\dots$ .

The $n\text{th}$ term of the arithmetic sequence is $a_n=a_1+\left(n-1\right)d$ .

Apply it,

  $\begin{array}{c}34=100+\left(12-1\right)d\\ 34=100+11d\\ 11d=-66\\ d=-6\end{array}$

Thus, the common difference for the arithmetic sequence is $d=-6$ .