Question

Prove that the (n+1) term of a G.P of which the first term is a and third term is b,is equal to the (2n+1) term of a G.P of which the first term is a and fifth term is b.

Answer 1

Hello,

$u_{1} =a\\ u_{3} =b\\ q= (\dfrac { u_{3} } {u_{1 } } )^ \frac{1}{2} \\\\\\ =(\dfrac{b}{a}) ^ \frac{1}{2} \\ u_{n+1} =a*q^{n}=a* (\dfrac{b}{a}) ^ \dfrac{n}{2} \\ \\\\\\\\\\\\v_{1} =a\\ v_{5} =b\\ r= (\dfrac { v_{5} } {v_{1 } } )^ \frac{1}{4} \\\\\\ v_{2n+1} =a*r^{2n}\\\\ =a* (\dfrac{b}{a}) ^ \dfrac{2n}{4}\\\\ =a* (\dfrac{b}{a}) ^ \dfrac{n}{2}$