Question

In a sequence the sum of n terms sn is given by sn=2n2+3n+1 for all values of n.show that sequence is an ap

Answer 1

I think there is a mistake in question its like that,

We have to prove that given sequence is not an AP.

Given---> Sₙ = 2n² + 3n + 1

To show---> Given sequence is not an AP.

Proof---> ATQ,

Sₙ = 2n² + 3n + 1

We put n = 1 in it

S₁ = 2 ( 1 )² + 3 ( 1 ) + 1

= 2 ( 1 ) + 3 + 1

= 2 + 4

S₁ = 6

S₁ means sum of one term means S₁ is equal to first term

a₁ = S₁ = 6

Putting n = 2 in Sₙ , we get

S₂ = 2 ( 2 )² + 3 ( 2 ) + 1

= 2 ( 4 ) + 6 + 1

= 8 + 7

= 15

S₂ means sum of first two terms i.e. sum of a₁ and a₂.

a₁ + a₂ = 15

6 + a₂ = 15

a₂ = 15 - 6

a₂ = 9

Now we put n = 3 , in Sₙ , we get,

S₃ = 2 ( 3 )² + 3 ( 3 ) + 1

= 2 ( 9 ) + 9 + 1

= 18 + 10

= 28

S₃ means sum of first three terms i.e. sum of a₁, a₂, a₃

a₁ + a₂ + a₃ = 28

=> 6 + 9 + a₃ = 28

=> 15 + a₃ = 28

=> a₃ = 28 - 15

=> a₃ = 13

So sequence is 6 , 9 , 13 , .................

a₂ - a₁ = 9 - 6 = 3

a₃ - a₂ = 13 - 9 = 4

Clearly, a₂ - a₁ ≠ a₃ - a₂

So given sequence is not an AP.