Question

If the 2nd,5th and 9th terms of non-constant a.p are in g.p, the common ratio of this g.p is?

Answer 1

since 2nd, 5th and 9th terms of non constant A P are in G P. 
So, in A.P.
2nd term = a+d,
5th term = a+4d,
9th term = a+8d
And G.P.
a, ar, ar∧2
a+d = a
a+4d = ar
a+8d = ar∧2
a+4d-(a-d) =a(r-1)
a+8d-(a+4d) = ar(r-1)
3d/4d = 1/r
common ratio of G.P. = 4/3

Answer 2

Answer:

Second term of an AP can be written as: a + d

Fifth term of an AP can be written as : a + 4d

Ninth term of an AP can be written as: a + 8d

General term of an AP = a + ( n - 1 ) d

According to the question, it is given that the above terms are in GP. We know the relation that:

⇒ b = √ac  [ Geometric Mean ]

⇒ b² = ac  

Now considering 2nd term as a, 5th term as b and 9th term as c, we get:

⇒ ( a + 4d )² = ( a + d ) ( a + 8d )

⇒ ( a² + 8ad + 16d² ) = ( a² + 8ad + ad + 8d² )

Bringing all the 'a' terms on LHS and 'd' terms to the RHS, we get:

⇒ ( a² - a² + 8ad - 8ad - ad ) = 8d² - 16d²

⇒ - ad = -8d²

⇒ ad = 8d(d)

Cancelling out 'd' from both the sides we get:

⇒ a = 8d

Now we know the relation between common ratio. It is the ratio between 2nd term, 5th term and 9th term. Substituting the values we get:

⇒ 2nd term = 8d + d = 9d

⇒ 5th term = 8d + 4d = 12d

⇒ 9th term = 8d + 8d = 16d

Therefore new GP = 9d, 12d, 16d

Common ratio between the above terms is given as:

⇒ r = Second term ÷ First Term

⇒ r = 12d/9d

⇒ r = 4/3

4/3 is the common ratio of the GP.