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# 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

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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

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1. We have,

Sn​=4n−n2

Taking n=1 and we get,

S1​=4×1−12

=4−1

S1​=3

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

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

So, first term a=3.

Now, taking n=2 and we get,

Sn​=4n−n2

S2​=4×2−22

S2​=4

Then, sum of first 2 terms is 4.

Now, finding Secondterm.

Then,

According to given question,

Firstterm+Secondterm=4

3+a2​=4

a2​=1

Hence the A.P.is

First term a=3

Common difference

d=Secondterm-firstterm

d=1-3

=-2

The third term of

a3​=a+2d

a3​=3+2×(−2)

a3​=−1

Similarly, 10th term

a10​=a+9d

=3+9(−2)

=3−18

=−15

Then,

nth term

an​=a+(n−1)d

an​=3+(n−1)(−2)

an​=3−2n+2

an​=5−2n

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