Chapter 11, Problem 4PT

To determine

To find the four arithmetic means between

  $-9and11$

Answer to Problem 4PT

The four arithmetic means between -9 and 11 are -5, -1, 3, 7.

Explanation of Solution

Given:

  $-9and11$

Concept Used:

  $n^{th}$ term of an arithmetic sequence with first term a and common term d is: $a_n=a_{\left(n-1\right)}+d$ .

Calculation:

In order to find four arithmetic means between

  $-9and11$

There are 6 terms in an arithmetic sequence.

The first term of the sequence $a_1=-9$

6^th terms of the sequence $a_6=11$

And $n=6$

To find 4 arithmetic means between $-9and11$ , first find d.

Use the formula for $n^{th}$ term of an arithmetic sequence with first term a and common term d is: $a_n=a+\left(n-1\right)d$ .

Here, given that first term is -9 and common difference is

  $\begin{array}{c}{a}_6=a_1+\left(6-1\right)d\\ 11=-9+5d\\ 11+9=5d\\ 5d=20\\ d=4\end{array}$

To find next four terms of given sequence find $a_2,a_3,a_4,a_5$ , as

  $\begin{array}{c}{a}_2=a_{2-1}+\left(4\right)\\ =a_1+4\\ =-9+4\\ =-5,\\ a_3=a_{3-1}+\left(4\right)\\ =a_2+4\\ =-5+4\\ =-1\\ a_4=a_{4-1}+\left(4\right)\\ =a_3+4\\ =-1+4\\ =3,\text{ and }\\ a_5=a_{5-1}+\left(4\right)\\ =a_4+4\\ =3+4\\ =7\end{array}$

Thus, the four arithmetic means between -9 and 11 are -5, -1, 3, 7.