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Rajan@2021
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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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