The ratio of sum to n terms of two A.P. is for every n ∈ N. Find the ratio of their 7th terms and mth terms.

Question

Philoid · Accepted Answer

Let the sum of n terms of the first A.P be:

⇒ S_n = × (2a + (n – 1)d) …………… (1)

Let the sum of n terms of the second A.P be:

⇒ S’_n = × (2a’ + (n – 1)d’) …………… (2)

Now according to the question:

⇒ =

Let’s consider the ratio these two AP’s m^th terms as:

T_m : T’_m

Now, recall that, nth term of an AP is T_n = a + (n – 1)d

⇒ T_m = a + (m – 1)d

⇒ T’^m = a’ + (m – 1)d’

Hence the ratio of these two AP’s m^th terms become:

⇒ =

On multiplying by 2, we get,

⇒ =

Now from the above formula of the ratio of mth terms of 2 Aps, we can find the ratio of 7^th terms of both Aps

So we have, =

⇒ =

∴ the ratio of m^th terms of the given 2 Aps is, 16m – 7 : 14m – 4

∴ the ratio of 7^th terms of the given 2 Aps is, 105 : 94

The ratio of sum to n terms of two A.P. is for every n ∈ N. Find the ratio of their 7^th terms and m^th terms.