Question

Insert $6$ geometric means between $14$ and $-\frac{7}{64}$.

Answer 1

Let $a=14,b=\frac{-7}{64}$
$6$ G.M between $a$ and $b$ are given by
$ar,a{r}^2,a{r}^3,a{r}^4,a{r}^5,a{r}^6$
$b=a{r}^7=\frac{-7}{64}$
$b=r^7=\frac{-1}{2^7}$
$r=\frac{-1}{2}$
Thus, the $6$ G.M between $a$ and $b$ are
$14\times \frac{-1}{2},14\times \frac{1}{4},14\times \frac{-1}{8},14\times \frac{1}{16},14\times \frac{-1}{32},14\times \frac{1}{64}$
i.e. $-7,\frac{7}{2},\frac{-7}{4},........\frac{7}{32}$