The ratio of the sum of m terms to sum of n terms of an A.P. is. Find the ratio of its mth term to its nth term.

As we know that sum of n terms of an A.P is:

⇒ S_n= × (2a + (n – 1)d)

So, sum of m terms of an A.P is:

⇒ S_m = × (2a + (m – 1)d)

From the question we know that,

⇒ =

⇒ =

⇒ =

⇒ =

⇒ m × (2a + (n – 1)d) = n × (2a + (m – 1)d)

⇒ 2ma + m(n – 1)d = 2na + n(m – 1)d

⇒ 2ma + md(n – 1) = 2na + nd(m – 1)

⇒ 2ma + mnd – md = 2na + mnd – nd

⇒ 2ma – md = 2na – nd

⇒ 2ma – 2na = md – nd

⇒ 2a(m – n) = d(m – n)

⇒ 2a = d

Now, m^th term of the given A.P is T_m = a + (m – 1)d

Now, n^th term of the given A.P is T_n = a + (n – 1)d

Now ratio between these 2 terms is:

⇒ =

Now we have, 2a = d

So we get,

⇒ =

⇒ =

⇒ =

⇒ =

⇒ =

∴ the ratio of m^th term of the given A.P to its n^th Term is:

T_m : T_n = 2m – 1 : 2n – 1