Chapter 12.1, Problem 10CFU

To determine

To calculate: The value of $n$ for the arithmetic sequence for which last term is $37$ , first term is $-13$ and common difference is $5$ .

Answer to Problem 10CFU

The value of $n$ for the arithmetic sequence is $11$ .

Explanation of Solution

Given information:

The arithmetic sequence has last term as $a_n=37$ , first term as $a_1=-13$ and common difference as $d=5$ .

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 last term as $a_n=37$ , first term as $a_1=-13$ and common difference as $d=5$ .

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,

  $\begin{array}{c}37=-13+\left(n-1\right)\left(5\right)\\ 37=-13+5n-5\\ 5n=55\\ n=11\end{array}$

Thus, the value of $n$ for the arithmetic sequence is $11$ .