If the sum of the first n terms of an AP is 4n – n², what is the first term (that is S₁)? What is the sum of first two terms? What is the second term? Similarly, find the 3rd, the 10th and the nth terms.

Given,

S_n = 4n - n²

First term = S₁

= 4(1) - 1²

= 4 - 1 = 3.

S₂ = 4(2) - 2²

= 8 - 4

= 4.

Second term = S₂ - S₁

= 4 - 3 = 1.

Third term = S₃ - S₂

= [4(3) - 3²] - 4

= [12 - 9] - 4

= -1.

Tenth term = S₁₀ - S₉

= [4(10) - 10²] - [4(9) - 9²]

= [40 - 100] - [36 - 81]

= [-60] - [-45]

= -60 + 45

= -15.

nth term = S_n - S_{n - 1}

= [4n - n²] - [4(n - 1) - (n - 1)²]

= [4n - n²] - [4n - 4 - (n² + 1 - 2n)]

= [4n - n²] - [4n - 4 - n² - 1 + 2n]

= 4n - 4n - n² + n² + 4 + 1 - 2n

= 5 - 2n.

Hence, first term = 3, sum upto 2 terms = 4, second term = 1, third term = -1, tenth term = -15 and nth term = 5 - 2n.