Question

Insert $19$ arithmetic means between $\frac{1}{4}$ and $-9\frac{3}{4}$.

Answer 1

The sequence of arithmetic means will form an A.P

with $a=\frac{1}{4}$ and $a_{21}=-9\frac{3}{4}=-\frac{39}{4}$

$t_n=a+(n-1)d\Rightarrow t_{21}=a+(21-1)d\Rightarrow -\frac{39}{4}=\frac{1}{4}+20d\Rightarrow d=-\frac{1}{2}$

So the airhmetic means are

$a+d,a+2d+a+3d...........a+19d\Rightarrow -\frac{1}{4},-\frac{3}{4},-\mathrm{1...........}-\frac{37}{4}$