The second, fifth and eleventh terms of an A.P are in a G.P and the seventh term is 4. Find (a) first term of an A.P. (b) common difference of an A.P. (c) common ratio of the G.P?

Let the first term of the A.P be a and the common difference be d.

The second term of the A.P is a + d, the fifth term is a + 4d, and the eleventh term is a + 10d.

Given that the second, fifth, and eleventh terms are in a G.P, we have:

(a + d)^2 = (a + 4d)(a + 10d)

Expanding both sides, we get:

a^2 + 2ad + d^2 = a^2 + 10ad + 4ad + 40d^2

Simplifying, we get:

2ad + d^2 = 14ad + 40d^2

Rearranging terms, we get:

12ad = 39d^2

Dividing by d on both sides, we get:

12a = 39d

Now, we are given that the seventh term of the A.P is 4.

So, a + 6d = 4

Substitute the value of 12a from the previous equation into this equation:

12(4/39) + 6d = 4

48/39 + 6d = 4

48/39 + 6d = 156/39

6d = 108/39

d = 18/39

d = 6/13

Now, substitute the value of d back into the equation a + 6d = 4:

a + 6(6/13) = 4

a + 36/13 = 4

a = 52/13 - 36/13

a = 16/13

(a) The first term of the A.P is 16/13.
(b) The common difference of the A.P is 6/13.
(c) To find the common ratio of the G.P, we can use the second and fifth terms of the A.P:

(a + d) / a = (a + 4d) / (a + d)

(16/13 + 6/13) / (16/13) = (16/13 + 24/13) / (16/13 + 6/13)

22/13 / 16/13 = 40/13 / 22/13

22/16 = 40/22

11/8 = 20/11

11/8 = 11/8

Therefore, the common ratio of the G.P is 11/8.

Let the first term of the A.P. be 'a' and the common difference be 'd'.

Therefore, the second term = a + d
Fifth term = a + 4d
Eleventh term = a + 10d

Given: The second, fifth, and eleventh terms are in G.P.
Therefore,
$$(a+d)(a+4d)=(a+10d)^2$$
$$\Rightarrow a^2+5ad+4d^2=a^2+20ad+100d^2$$
$$\Rightarrow -15ad+96d^2=0$$
$$\Rightarrow -5ad+32d=0$$
$$\Rightarrow a=\frac{32d}{5}.............(i)$$

It is also given that the seventh term is 4.
Therefore,
$$a+6d=4$$
Substituting the value of 'a' from (i), we get
$$\frac{32d}{5}+6d=4$$
$$\Rightarrow 32d+30d=20$$
$$\Rightarrow 62d=20$$
$$\Rightarrow d=\frac{20}{62}=\frac{10}{31}$$

Substituting the value of 'd' in (i), we get
$$a=\frac{32}{5}\times \frac{10}{31}=\frac{320}{155}$$

Therefore,
(a) First term of the A.P. = $$a=\frac{320}{155}$$
(b) Common difference of the A.P. = $$d=\frac{10}{31}$$
(c) Common ratio of the G.P. = $$\frac{a+4d}{a+d}=\frac{\frac{320}{155}+4\times \frac{10}{31}}{\frac{320}{155}+\frac{10}{31}}=\frac{\frac{320+200}{155}}{\frac{320+50}{155}}=\frac{520/155}{370/155}=\frac{520}{370}=\frac{26}{185/9}$$