Chapter 12.1, Problem 32E

To determine

To calculate: The first term for the arithmetic sequence for which common difference is $\frac{1}{3}$ , and seventh term is $10\frac{2}{3}$ .

Answer to Problem 32E

The first term for the arithmetic sequence is $a_1=\frac{26}{3}$ .

Explanation of Solution

Given information:

The arithmetic sequence has common difference as $d=\frac{1}{3}$ , and seventh term as $a_7=10\frac{2}{3}$ .

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 common difference as $d=\frac{1}{3}$ , and seventh term as $a_7=10\frac{2}{3}$ .

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,

  $\begin{array}{c}10\frac{2}{3}=a_1+\left(7-1\right)\left(\frac{1}{3}\right)\\ \frac{32}{3}=a_1+2\\ a_1=\frac{32}{3}-2\\ a_1=\frac{26}{3}\end{array}$

Thus, the first term for the arithmetic sequence is $a_1=\frac{26}{3}$ .