**Q) **The ratio of the 11^{th} term to 17^{th} term of an A.P. is 3:4. Find the ratio of 5^{th} term to 21^{st} term of the same A.P. Also, find the ratio of the sum of first 5 terms to that of first 21 terms.

**Ans: **We know that n^{th} term of an A.P. = a + (n-1) d

Therefore, 11^{th} term, N_{11} = a + 10d and 17^{th} term, N_{17 }= a + 16d

Given that : =

=

or 4a + 40 d = 3a + 48 d

or a = 8d ……………………………………. equation no. (i)

Now 5^{th} term N_{5} = a + 4d and 21^{st} term N_{21} = a + 20 d

Therefore, =

Substituting a = 8d from equation (i), we get

= =

**Therefore, N _{5}:N_{21 }= 3:7**

Let’s calculate ratio of the sum of first 5 terms to that of first 21 terms.

We know that sum of n terms of an A.P. S_{n }= (2a + (n-1) d)

Sum of first 5 terms, S_{5} = (2a + 4d)

and Sum of first 21 terms, S_{21} = (2a + 20d)

Therefore, =

=

Substituting a = 8d from equation (i), we get

=

=

=

**Therefore, S _{5 }: S_{21 }=**

**25 : 189**