If the sum of the first n terms of an AP is $4n – n^2 $ , what is the first term (that is $S_1$ )? What is the sum of first two terms? What is the second term? Similarly, find the $3^{rd}$ , the $10^{th}$ and the $n^{th}$ terms.

Question

If the sum of the first n terms of an AP is  $4n – n^2 $ , what is the first term (that is  $S_1$ )? What is the sum of first two terms? What is the second term? Similarly, find the  $3^{rd}$ , the  $10^{th}$  and the  $n^{th}$  terms.

cdquestions Admin · Accepted Answer

Given that, $S_n = 4n − n^2$ First term,  $a_1 = S_1 = 4(1) − (1)^2 = 4 − 1 = 3$ Sum of first two terms,  $S_2 = 4(2) − (2)^2 = 8 − 4 = 4$ Second term,  $a_2 = S_2 − S_1 = 4 − 3 = $ 1 $d = a_2 − a_1 = 1 − 3 = −2$ $a_n = a + (n − 1)d $ $a_n = 3 + (n − 1) (−2)$ $a_n  = 3 − 2n + 2$ $a_n = 5 − 2n$ Therefore, $a_3 = 5 − 2(3) = 5 − 6 = −1$ $a_{10} = 5 − 2(10) = 5 − 20 = −15$ Hence, the sum of first two terms is 4. The second term is 1. 3 rd , 10 th , and n th terms are −1, −15, and 5−2n respectively.

If the sum of the first n terms of an AP is \(4n – n^2 \), what is the first term (that is \(S_1\))? What is the sum of first two terms? What is the second term? Similarly, find the \(3^{rd}\), the \(10^{th}\) and the \(n^{th}\) terms.

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