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, N₁₁ = a + 10d and 17^th term, N₁₇ = a + 16d

Given that : =

    =

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

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

Now 5^th term N₅ = a + 4d and 21^st term N₂₁ = a + 20 d

Therefore,  =

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

   = =

Therefore, N₅:N₂₁ = 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, S₅ = (2a + 4d)

and Sum of first 21 terms, S₂₁ =    (2a + 20d)

Therefore, =

=

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

=

=

=

Therefore, S₅ : S₂₁ = 25 : 189