Question

if k,2k+2,3k+3,......are G.P then find common ratio of G.P

Answer 1

its easy see

2k+2/k = 3k+3/2k+2. as it is a gp

so

2k+2×2k+2 = 3k^2 + 3k

4k^2 + 8k + 4 = 3k^2 + 3k

k^2 + 5k +4 =0

so k= -1 or -4

so ratio is easy to find keep values of k and take any two consecutive terms in ratio

Answer 2

${b}^{2} = ab \\ {(2k + 2)}^{2} = k(3k + 3) \\ 4 {x}^{2} + 8x + 4 = 3 {x }^{2} + 3x \\ {x }^{2} + 5x + 4 = 0 \\ x = - 1 \: \: or - 4$
GP is -1,0,0
this is now possible
-4,-6,-9
hence ratio is 2/3