Question

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

Answer 1

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.

⇒ $r = \frac{a6}{a2} = \frac{a18}{a6}$

or, $r = \frac{a + 5d}{a+d} = \frac{a+ 17d}{a+ 5d}$

By cross multiplying, we get

$(a+5d)^{2} = (a+d)(a+17d)$

or, $a^{2}+ 25d^{2} + 10ad = a^{2} + 17ad+ ad+ 17d^{2}$

Now, simplifying the above expression, we get that

$8d^{2} = 8ad\\or, a = d$

or, r = a = d

Hence, the Common Ratio = First Term = Common Difference.