Q. 15

# The n^{th} term of a sequence is given by a_{n} = 2n^{2} + n+ 1. Show that it is not an A.P.

Answer :

Given,

a_{n} = 2n^{2} + n + 1

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

When n = 1:

a_{1} = 2(1)^{2} + 1 + 1

⇒ a_{1} = 2(1) + 2

⇒ a_{1} = 2 + 2

⇒ a_{1} = 4

When n = 2:

a_{2} = 2(2)^{2} + 2 + 1

⇒ a_{2} = 2(4) + 3

⇒ a_{2} = 8 + 3

⇒ a_{2} = 11

When n = 3:

a_{3} = 2(3)^{2} + 3 + 1

⇒ a_{3} = 2(9) + 4

⇒ a_{3} = 18 + 4

⇒ a_{3} = 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

a_{1} = 4, a_{2} = 11, a_{3} = 22

Now, a_{2} – a_{1} = 11 – 4 = 7

a_{3} – a_{2} = 22 – 11 = 11

As a_{2} – a_{1} is not equal to a_{3} – a_{2}

The given sequence is not an A.P.

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