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 3rd, the 10th and the nth terms.

Solution:

Given that,

S_n = 4n − n²

First term, a = S ₁ = 4(1) − (1)² = 4 − 1 = 3

Sum of first two terms = S₂= 4(2) − (2)² = 8 − 4 = 4

Second term, a₂ = S₂ − S₁ = 4 − 3 = 1

Common difference, d = a₂− a = 1 − 3 = −2

N^th term, a_n = a + (n − 1) d

= 3 + (n − 1)(−2)

= 3 − 2n + 2

= 5 − 2n

Therefore, a₃ = 5 − 2(3) = 5 – 6 = −1

a₁₀ = 5 − 2(10) = 5 − 20 = −15

Hence, the sum of the first two terms is 4. The second term is 1.

The 3^rd, the 10^th, and the n^th terms are −1, −15, and 5 − 2n respectively.