In an A.P ratio of 3rd term and 5th term is 3:7.

a)Find the ratio of the 4th and 6th term?
b)Find the ratio of the 6th and 12th term?

1 a

2 a+d
3 a+2d
4 a+3d
5 a+4d

n a + (n-1)d

(a+2d)/(a+4d) = 3/7

3 a + 12d = 7a + 14 d
4a=-2d
d = -2a

fourth/sixth (a+3d)/(a+5d)
= (a-6a)/(a-10a)
= -5a/-9a
= 5/9
etc

To find the ratios of the terms in an arithmetic progression (A.P.), we need to know the common difference (d). However, in this case, we are only given the ratios between the terms.

To solve these questions, let's break the problem down step by step:

a) Finding the ratio of the 4th and 6th term:
Let's assume that the common difference is "d".

Given that the ratio of the 3rd term to the 5th term is 3:7, we can express these terms as follows:
3rd term = (1st term + 2d)
5th term = (1st term + 4d)

Now let's set up the equation:
(1st term + 2d)/(1st term + 4d) = 3/7

To eliminate the denominators, let's cross-multiply:
(1st term + 2d)*7 = (1st term + 4d)*3

Expand the equation:
7(1st term) + 14d = 3(1st term) + 12d

Simplify:
7(1st term) - 3(1st term) = 12d - 14d
4(1st term) = -2d
(1st term) = -2d/4
(1st term) = -d/2

Now, we can find the 4th term and the 6th term using this information:
4th term = (1st term + 3d) = (-d/2 + 3d) = (5d/2)
6th term = (1st term + 5d) = (-d/2 + 5d) = (9d/2)

Finally, we can express the ratio of the 4th and 6th term:
4th term/6th term = (5d/2)/(9d/2) = 5d/9d

Simplify the ratio:
4th term/6th term = 5/9

Therefore, the ratio of the 4th and 6th term is 5:9.

b) Finding the ratio of the 6th and 12th term:
Now, let's find the 12th term in terms of the first term and the common difference "d".
12th term = (1st term + 11d)

To find the ratio of the 6th and 12th term, we can express the terms as follows:
6th term = (1st term + 5d)
12th term = (1st term + 11d)

We can now set up the ratio:
6th term/12th term = (1st term + 5d)/(1st term + 11d)

Simplify the ratio:
6th term/12th term = (1 + 5d)/(1 + 11d)

Therefore, the ratio of the 6th and 12th term is (1 + 5d):(1 + 11d).