Geometric Sequences

Let the first term be $a$ and the common difference be $r$ . The $n$ -th term of the progression is given as

$t_n = ar^{n-1}$

Now, we know that

$t_2 = ar = -6\\ t_5 = ar^4 = 162$

Now, we divide these two equations:

$\dfrac{t_5}{t_2} = \dfrac{ar^4}{ar} = \dfrac{162}{-6}\\ \implies r^3 = -27\\ r=-3$

Then,

$a(-3) = -6\\ a=2$

Therefore, $t_n = \boxed{2(-3)^{n-1}}$

$t2 = t1(r)$ ------ (1)

$t5 = t1(r^4)$ ------ (2)

$[t1(r^4)]/[t1(r)] = 162/-6$ ------ (2) / (1)

$r^3 = -27$

r = -3

t1(-3) = -6

t1 = 2

$tn = 2*(-3)^{n-1}$