Question

in a geometric progression,the sum of the second and third term is 6,and the sum of the third and fourth terms is -12.find the first term and the common ratio

Answer 1

Assuming the terms of the GP to be \[a\text{, }ar\text{, }ar^2\text{, }ar^3\]
We can say \[ar+ar^2 = 6 \text{ and } ar^2+ar^3 = -12\]
So we have \[ar(1+r) = 6 \text{ and } ar^2(1+r) = -12\]
Dividing, we get \[r=-2\]
Substituting r in anyone equation, we have \[a = 3\]