Whats the missing term in the geometric sequence 25, __, 225?

Nov 13, 2015

Depending on the common ratio ($r = \pm 3$), either ${A}_{2} = 75$ or ${A}_{2} = - 75$.

Explanation:

${A}_{1} = 25$

${A}_{3} = 225$

${A}_{n} = {A}_{1} {r}^{n - 1}$

$\implies {A}_{3} = {A}_{1} {r}^{3 - 1}$

$\implies 225 = 25 {r}^{2}$

$\implies 9 = {r}^{2}$

$\implies r = - 3 , r = 3$

If $r = - 3$

${A}_{2} = 25 {\left(- 3\right)}^{2 - 1}$

$\implies {A}_{2} = 25 \left(- 3\right)$

$\implies {A}_{2} = - 75$

If $r = 3$

${A}_{2} = 25 {\left(3\right)}^{2 - 1}$
$\implies {A}_{2} = 25 \left(3\right)$

$\implies {A}_{2} = 75$