The ratio of the 11th term of 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.

Given: a₁₁ : a₁₈ = 2:3

To find: a₅:a₂₁

S₅:S₂₁

Formula Used:

Sum of “n” terms of an AP:

Where S_n is the sum of first n terms

nth term of an AP is:

a_{n =} a + (n - 1) d

Where a_n is the last term.

n = no of terms

a = first term

d = common difference

Explanation:

Let “a” and “d” the first term and common difference of an AP.

a₁₁:a₁₈ = 2:3

a + 10d : a + 17d = 2:3

3a + 30d = 2a + 34d

a = 4d [ eqn 1]

a₅ = a + 4d = 4d + 4d = 8d [ by eqn 1] [ Eqn 2]

a₂₁ = a + 20d = 4d + 20d = 24d [ by eqn 1] [ Eqn 3]

now using the formula

= 30d

= 21( 14 )d

= 294

S₅:S₂₁ = 30 : 294 = 5: 49