It is given that if the sum of the first n terms of an Ap is 4n-n2 what is first term? what is the sum of first two terms? what is the second term? similarly find the 3rd and 10th and nth terms.

Sir please give me a detailed solution of this question as it is very important for board examination and it is from ML Aggarwal Avichal Publication Arithmetic Progression Chapter 9 Question no 17

We have,

Snâ€‹=4nâˆ’n2

TakingÂ n=1Â and we get,

S1â€‹=4Ã—1âˆ’12

=4âˆ’1

S1â€‹=3

So, the sum of fristr term of an A.P. isÂ 3.

But, sum of first term will be the first term.

So, first termÂ a=3.

Now, takingÂ n=2Â and we get,

Snâ€‹=4nâˆ’n2

S2â€‹=4Ã—2âˆ’22

S2â€‹=4

Then, sum of firstÂ 2Â terms is 4.

Now, findingÂ Secondterm.

Then,

According to given question,

Firstterm+Secondterm=4

3+a2â€‹=4

a2â€‹=1

Hence theÂ A.P.is

First termÂ a=3

Common difference

d=Secondterm-firstterm

d=1-3

=-2

The third term of

a3â€‹=a+2d

a3â€‹=3+2Ã—(âˆ’2)

a3â€‹=âˆ’1

Similarly,Â 10th term

a10â€‹=a+9d

=3+9(âˆ’2)

=3âˆ’18

=âˆ’15

Then,

nth term

anâ€‹=a+(nâˆ’1)d

anâ€‹=3+(nâˆ’1)(âˆ’2)

anâ€‹=3âˆ’2n+2

anâ€‹=5âˆ’2n