If 2nd, 3rd and 6th terms of an AP are the three consecutive terms of

1.	If 2nd, 3rd and 6th terms of an AP are the three consecutive terms of a GP then find the common ratio of the GP.
Answer» We have been given that 2^nd, 3^rd and 6^th terms of an AP are the three consecutive terms of a GP. Let the three consecutive terms of the G.P. be a,ar,ar² . Where a is the first consecutive term and r is the common ratio. 2^nd, 3^rd terms of the A.P. are a and ar respectively as per the question. ∴ The common difference of the A.P. = ar - a And the sixth term of the A.P. = ar² Since the second term is a and the sixth term is ar² (In A.P.) We use the formula: t = a + (n - 1)d ∴ ar² = a + 4(ar - a)…(the difference between 2^nd and 6^th term is 4(ar - a)) ⇒ ar² = a + 4ar - 4a ⇒ ar² + 3a - 4ar = 0 ⇒ a(r² - 4r + 3) = 0 ⇒ a(r - 1)(r - 3) = 0 Here, we have 3 possible options: 1) a = 0 which is not expected because all the terms of A.P. and G.P. will be 0. 2) r = 1,which is also not expected because all th terms would be equal to first term. 3) r = 3,which is the required answer. Hence, Common ratio = 3

If 2nd, 3rd and 6th terms of an AP are the three consecutive terms of a GP then find the common ratio of the GP.

Answer»

We have been given that 2^nd, 3^rd and 6^th terms of an AP are the three consecutive terms of a GP.

Let the three consecutive terms of the G.P. be a,ar,ar² .

Where a is the first consecutive term and r is the common ratio.

2^nd, 3^rd terms of the A.P. are a and ar respectively as per the question.

∴ The common difference of the A.P. = ar - a

And the sixth term of the A.P. = ar²

Since the second term is a and the sixth term is ar² (In A.P.)

We use the formula: t = a + (n - 1)d

∴ ar² = a + 4(ar - a)…(the difference between 2^nd and 6^th term is 4(ar - a))

⇒ ar² = a + 4ar - 4a

⇒ ar² + 3a - 4ar = 0

⇒ a(r² - 4r + 3) = 0

⇒ a(r - 1)(r - 3) = 0

Here, we have 3 possible options:

1) a = 0 which is not expected because all the terms of A.P. and G.P. will be 0.

2) r = 1,which is also not expected because all th terms would be equal to first term.