If the first,second and fifth terms of AP are three consecutive terms of a GP,find the common ratio?

the answer is 3

answer is 3

To find the common ratio (r) of a geometric progression (GP), we need to carefully analyze the given information.

Let's denote the first term of the arithmetic progression (AP) as 'a', and the common difference as 'd'. Then, the terms of the AP would be as follows:

First term: a
Second term: a + d
Fifth term: a + 4d

Now, we are given that these three terms of the AP (a, a + d, a + 4d) are also consecutive terms of a geometric progression (GP). For a GP, each term is obtained by multiplying the previous term by a constant ratio 'r'. Hence, we can write:

Common ratio of the GP: r

Now, let's examine the relation between the terms of the GP:

First term: a
Second term: (a + d) * r
Fifth term: (a + 4d) * r^4 (since the common ratio is multiplied 4 times)

According to the given conditions, these three terms of the GP are the same as the corresponding terms of the AP. Therefore, we can equate them:

a = a
(a + d) * r = a + d
(a + 4d) * r^4 = a + 4d

Let's solve these equations step by step to determine the common ratio (r):

Equation 1: a = a
This equation is trivially true and does not provide any additional information.

Equation 2: (a + d) * r = a + d
Expanding this equation:
ar + dr = a + d
Factoring out the common term 'd':
d(r - 1) = a - ar
Dividing both sides by 'd' (remembering that d ≠ 0):
r - 1 = 1 - r
Adding 'r' to both sides:
r - 1 + r = 1 - r + r
2r - 1 = 1
Adding '1' to both sides:
2r - 1 + 1 = 1 + 1
2r = 2
Dividing by '2':
r = 1

Equation 3: (a + 4d) * r^4 = a + 4d
Substituting r = 1 into the equation:
(a + 4d) * 1^4 = a + 4d
a + 4d = a + 4d

This equation is also trivially true, indicating that any value of 'a' and 'd' would satisfy the given conditions.

Therefore, the common ratio (r) of the GP is 1.

for the AP

first term = a
2nd term = a+d
fifth term = a+4d

but these also form a GP, so
(a+d)/a = (a+4d)/(a+d)
a^2 + 2ad + d^2 = a^2 + 5ad
3ad = d^2
d = 3a

common ratio
= (a+d)/a
= (a+3a)/a = 4