Question

if a gp is made by taking terms of ap.t1,t5,t9.find common ratio of gp

Answer 1

we know,
nth term (tn) in GP = a.r^(n-1) where a is first term and r is common ratio .

A/C to question,
t1 , t5 , t9 are in AP
so,
t5 - t1 = t9 - t5

ar^(5-1) - ar^(1-1) = ar^(9-1) -ar^(5-1)

ar⁴ - a = ar^8 - ar⁴

2ar⁴ = ar^8 + a

2r⁴ = r^8 + 1

(r⁴)² - 2r⁴ + 1 = 0

let r⁴ = P

P² -2P +1 = 0
(P -1)² = 0
P = 1
now put P = r⁴

r⁴ -1 = 0
(r² + 1)(r² -1) = 0

(r² + 1) ≠ 0 hence, r² = 1
so, r = -1 and +1