Let the numbers in G.P. be \(\frac{a}{r}\), a, ar … (i)
Product \(\frac{a}{r}\), a. ar = 1000
⇒ a3 = 1000
⇒ a = 10
According to question
A.P. a1 = \(\frac{a}{r}\) = \(\frac{10}{r}\)
a2 = a + 6 = 10 + 6 = 16
a3 = ar + 7 = 10r + 7
Also, a3 = a1 + 2(a2 − a1) [∴ a, b, c are in A.P.]
⇒ 10r + 7\(\frac{10}{r}\) + 2\(\big[10 − \frac{10}{r}\big]\)
⇒ 10r2 + 7r = 10 + 32r − 20
⇒ 10r2 − 25r + 10 = 0
⇒ (r − 2)(10r − 5) = 0
⇒ r = 2 or \(\frac{1}{2}\)
Substituting the value of a and r in eq (i), we get G.P. : 5,10, 20… when r = 2 and G.P. : 20,10, 5... when r = \(\frac{1}{2}\)