If second, sixth and eighteenth term of an AP are the consecutive term of GP. Find the commom ratio​

Answer:Common Ratio = First Term = Common Difference.Step-by-step explanation:The general term of AP is an = a + (n-1)dSo, the second term of AP is a2 = a + dThe sixth term of AP is a6 = a + 5dThe eighteenth term of AP is a18 = a + 17dNow, the terms a2, a6 and a18 are in GP.⇒ [tex]r = \frac{a6}{a2} = \frac{a18}{a6}[/tex]or, [tex]r = \frac{a + 5d}{a+d} = \frac{a+ 17d}{a+ 5d}[/tex]By cross multiplying, we get[tex](a+5d)^{2} = (a+d)(a+17d)[/tex]or, [tex]a^{2}+ 25d^{2} + 10ad = a^{2} + 17ad+ ad+ 17d^{2}[/tex]Now, simplifying the above expression, we get that[tex]8d^{2} = 8ad\\or, a = d[/tex]or, r = a = dHence, the Common Ratio = First Term = Common Difference.