Question

show that the sequence described by an = 1/3n + 1/6 is an A. P​

Answer 1

Rule:

The condition that the three terms x, y, z be in A.P. is

y - x = z - y

Step-by-step explanation:

The sequence is described by

aₙ = (1/3) n + 1/6

We put n = 1, 2, 3, ... to find the sequence. Then

a₁, a₂, a₃, ..., aₙ

is the sequence given.

Here a₁ = 1/3 + 1/6

a₂ = 2/3 + 1/6

a₃ = 3/3 + 1/6

Now, a₂ - a₁ = (2/3 + 1/6) - (1/3 + 1/6)

= 2/3 + 1/6 - 1/3 - 1/6

= 1/3

and a₃ - a₂ = (3/3 + 1/6) - (2/3 + 1/6)

= 3/3 + 1/6 - 2/3 - 1/6

= 1/3

∵ a₂ - a₁ = a₃ - a₂, the condition for AP is satisfied.

∴ the given sequence is in AP.

This completes the proof.