**Question:**

Insert six arithmetic means between 11 and -10

**Solution:**

To find: Six arithmetic means between 11 and -10

Formula used: (i) $d=\frac{b-a}{n+1}$, where, $d$ is the common difference

n is the number of arithmetic means

(ii) $A_{n}=a+n d$

We have 11 and -10

Using Formula, $d=\frac{b-a}{n+1}$

$d=\frac{-10-(11)}{6+1}$

$d=\frac{-21}{7}$

$d=-3$

Using Formula, $A_{n}=a+n d$

First arithmetic mean, $A_{1}=a+d$

$=11+(-3)$

$=8$

Second arithmetic mean, $\mathrm{A}_{2}=\mathrm{a}+2 \mathrm{~d}$

$=11+2(-3)$

$=11+(-6)$

$=5$

Third arithmetic mean, $\mathrm{A}_{3}=\mathrm{a}+3 \mathrm{~d}$

$=11+3(-3)$

$=11+(-9)$

$=2$

Fourth arithmetic mean, $\mathrm{A}_{4}=\mathrm{a}+4 \mathrm{~d}$

$=11+4(-3)$

$=11+(-12)$

$=-1$

Fifth arithmetic mean, $A_{5}=a+5 d$

$=11+5(-3)$

$=11+(-15)$

$=-4$

Sixth arithmetic mean, $A_{6}=a+6 d$

$=11+6(-3)$

$=11+(-18)$

$=-7$

Ans) The six arithmetic means between 11 and $-10$ are $8,5,2,-1,-4$ and $-7$.