Question

The ratio of the 11th term to the 18th term of an AP is `2:3`. Find the ratio of the 5th term to the 21st term and also the ratio of the sum of the first five terms to the sum of the first 21 terms.

Answer 1

Let a and d be the first term and common difference of an AP.
Given that, `a_(11) :a_(18)=2: 3`
`implies(a+10d)/(a+17d)=(2)/(3)`
`implies 3a +30d=2a+34d`
`impliesa =4d " "` ...(i)
Now, `a_(5)= a+4d=4d+4d=8d " "` [from Eq. (i)]
and`a_(21)= a+20d=4d+20d=24d " "` [from Eq. (i)]
` :. a_(5):a_(21)=8d:24d=1:3`
Now sum of the first five terms,
`S_(5)=(5)/(2)[2a+(5-1)d]`
` " " =(5)/(2)[2(4d)+4d] " "`[from Eq. (i)]
` " "=(5)/(2)[8d+4d]=(5)/(2)xx12d=30d`
and sum of the first 21 terms,
`S_(21)=(21)/(2)[2a+(21-1)d]`
` " " =(21)/(2)[2(4d)+20d] " "`[from Eq. (i)]
` " "=(21)/(2)(28d)=294d`
So, ratio of the sum of the first five terms to the sum of the first 21 terms
`S_(5):S_(21)=30d:294d=5:49`