Question

The n^th term of a sequence is given by a_n = 2n² + n+ 1. Show that it is not an A.P.

Answer 1

Given, 

a_n = 2n² + n + 1 

We can find first three terms of this sequence by putting values of n from 1 to 3. 

When n = 1 : 

a₁ = 2(1)² + 1 + 1 

⇒ a₁ = 2(1) + 2 

⇒ a₁ = 2 + 2 

⇒ a₁ = 4 

When n = 2 : 

a₂ = 2(2)² + 2 + 1 

⇒ a₂ = 2(4) + 3 

⇒ a₂ = 8 + 3 

⇒ a₂ = 11 

When n = 3 : 

a₃ = 2(3)² + 3 + 1 

⇒ a₃ = 2(9) + 4 

⇒ a₃ = 18 + 4 

⇒ a₃ = 22 

∴ First three terms of the sequence are 4, 11, 22. 

A.P is known for Arithmetic Progression whose common difference = a^_n – a_n-1 where n > 0 

a₁ = 4, a₂ = 11, a₃ = 22 

Now, 

a₂ – a₁ = 11 – 4 = 7 

a₃ – a₂ = 22 – 11 = 11 

As a₂ – a₁ is not equal to a₃ – a₂ 

The given sequence is not an A.P.