Answer:
Common Ratio = First Term = Common Difference.
Step-by-step explanation:
The general term of AP is an = a + (n-1)d
So, the second term of AP is a2 = a + d
The sixth term of AP is a6 = a + 5d
The eighteenth term of AP is a18 = a + 17d
Now, 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 = d
Hence, the Common Ratio = First Term = Common Difference.
