Question

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

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Answer 1

Common Ratio = First Term = Common Difference.

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 = (a6)/(a2) = (a18)/(a6)`

or,
`r = (a + 5d)/(a+d) = (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.