Chapter 4.7, Problem 22WE

To determine

To solve the equation $l=a+\left(n-1\right)d$ , for n .

Answer to Problem 22WE

  $n=\frac{l-a}{d}+1,d\ne 0$

Explanation of Solution

Given:

The equation, $l=a+\left(n-1\right)d$

Concept Used:

• To get rid of a number in addition from one side, subtract the same number from both sides of equal sign.
• To get rid of a number in subtraction from one side, add the same number both sides of equal sign.
• To get rid of a number in multiplication from one side, divide the same number from both sides of equal sign.
• To get rid of a number in division from one side, multiply the same number both sides of equal sign.

Calculation:

To solve the given equation $l=a+\left(n-1\right)d$ for n , isolate the variable term n on one side by performing some basic algebraic operations to get rid of the other numbers and terms associated with it.

Here to isolate n on right side, first subtract a from both sides and then simplify further by dividing d from both sides followed by adding 1 to both sides as shown below,

  $\begin{array}{c}l=a+\left(n-1\right)d\\ l-a=a+\left(n-1\right)d-a\\ l-a=\left(n-1\right)d\\ \frac{l-a}{d}=\frac{\left(n-1\right)d}{d}\\ \frac{l-a}{d}=n-1\\ \frac{l-a}{d}+1=n-1+1\\ \Rightarrow n=\frac{l-a}{d}+1\end{array}$

Thus, on solving for the indicated variable it gives $n=\frac{l-a}{d}+1$ .

Since it is a rational expression after solving for the indicated variable, so the denominator should not be zero. That is, $d\ne 0$ .