Correct Question:

The ratio of seventh to the THIRD term of the ap is 12 : 5. find the ratio of 13th term and 4th term of the same AP.

Answer:-

Given:

Ratio of 7th term and 3rd term = 12 : 5

Let the 7th term be 12x and 3rd term be 5x.

We know that,

nth term of an AP [a(n)] = a + (n - 1)d

Hence,

→ a(7) = a + (7 - 1)d

→ 12x = a + 6d -- equation (1)

Similarly,

→ a + 2d = 5x -- equation (2)

Subtract equation (2) from (1).

→ a + 6d - (a + 2d) = 12x - 5x

→ a + 6d - a - 2d = 7x

→ 4d = 7x

→ d = 7x/4

Substitute the value of "d" in equation (1).

→ a + 6(7x)/4 = 12x

→ a = 12x - 42x/4

→ a = (48x - 42x) / 4

→ a = 6x / 4

→ a = 3x/2

Hence,

a(13) = 3x/2 + (13 - 1)(7x/4)

→ a(13) = 3x/2 + 12(7x/4)

→ a(13) = (6x + 84x) / 4

→ a(13) = 90x/4

a(4) = 3x/2 + (3) * (7x/4)

→ a(4) = (6x + 21x)/4

→ a(4) = (27x)/4

Ratio of 13th and 4th terms = [ (90x)/4 ] / [ (27x)/4 ]

→ Ratio of 13th and 4th terms = (90x/4) * (4/27x)

→ Ratio of 13th and 4th terms = 10 / 3

→ Ratio of 13th and 4th terms = 10 : 3

Therefore, the ratio of 13th and 4th terms of the given AP is 10 : 3.