Let a and d be the first term and common difference of an AP.
Given that, a11:a18=2:3
⇒ a+10da+17d=23
⇒ 3a+30d=2a+34d
⇒ a=4d ...(i)
Now, a5 = a + 4d = 4d + 4d = 8d [from Eq.(i)]
And a21 = a + 20d = 4d + 20d = 24d [from Eq. (i)]
a5:a21 = 8d : 24d = 1 : 3
Now, sum of the first five terms, S5=52[2a+(5−1)d]
=52[2(4d)+4d] [from Eq.(i)]
=52(8d+4d)=52×12d=30d
And, sum of the first 21 terms, S21=212[2a+(21−1)d]
=212[2(4d)+20d] from Eq..(i)]
So, ratio of the sum of the first five terms to the sum of the first 21 terms is,
S5:S21=30d:294d=5:49