Question

Find an explicit rule for the nth term of a geometric sequence where the second and fifth terms are -6 and 162, respectively. Please explain this to me very carefully.

Answer 1

Hello,

$u_{0} =a\\ u_{1} =a*r\\ u_{2} =a*r^{2}=-6\\ u_{3} =a*r^{3}\\ u_{4} =a*r^{4}\\ u_{5} =a*r^{5}=162\\ ....\\ \boxed{ u_{n} =a*r^{n}} \\ \dfrac{ 162}{-6} = \dfrac{ u_{5} }{ u_{3}} = \dfrac{ a*r^{5}}{ a*r^{2}} =r^3= -27\\ ==\ \textgreater \ r=-3\\ u_{2} =a*r^{2}=-6=a*(-3)^2 \ ==\ \textgreater \ a=-\frac{6}{9} =-\frac{2}{3}\\ \boxed{ u_{n} =-\frac{2}{3}*(-3)^{n}=(-1)^{n+1}*2*3^{n-1}}$

Answer 2

The first step is to determine the equations used in the geometric sequences of the second and fifth terms. For the second term, the formula is -6=a*r². The equation for the fifth term is 162=a*r⁵. Therefore, the general formula is uₓ=a*rⁿ.