Question

Which of the following are APs? If they form an A.P. find the common difference d and write three more terms. `3, 3 + sqrt2, 3 + 2sqrt2, 3 + 3sqrt2`

Answer 1

`3, 3 + sqrt2, 3 + 2sqrt2, 3 + 3sqrt2`

Here,

a₂ - a₁ = `3 + sqrt2 - 3 = sqrt2`

a₃ - a₂ = `(3 + 2sqrt2) - (3 + sqrt2) = sqrt2`

a₄ - a₃ = `(3 + 3sqrt2) - (3 + 2sqrt2) = sqrt2`

⇒ a_n+1 - a_n is same every time.

Therefore, `d = sqrt2` and the given numbers are in A.P.

Three more terms are

a₅ = `(3 + sqrt2) + sqrt2 = 3 + 4sqrt2`

a₆ = `(3 + 4sqrt2) + sqrt2 = 3 + 5sqrt2`

a₇ = `(3 + 5sqrt2) + sqrt2 = 3 + 6sqrt2`