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)} \\ ( 162)/(-6) = ( u_(5) )/( u_(3)) = ( a*r^(5))/( a*r^(2)) =r^3= -27\\ ==\ \textgreater \ r=-3\\ u_(2) =a*r^(2)=-6=a*(-3)^2 \ ==\ \textgreater \ a=-(6)/(9) =-(2)/(3)\\ \boxed{ u_(n) =-(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.