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 3^rd, 10^th and the n^th terms.

Given, S_n = 4n - n²

Therefore,

 S₁ = 4 × 1 - 1² = 4 - 1 = 3

 S₂ = 4 × 2 - 2² = 8 - 4 = 4

 S₃ = 4 × 3 - 3² = 12 - 9 = 3

 a₂ = S₂ - S₁ = 4 - 3 = 1

 a₃ = S₃ - S₂ = 3 - 4 = - 1

Now,

a₁₀  = S₁₀ - S₉

= (4 × 10 - 10²) - (4 × 9 - 9²)

= (40 - 100) - (36 - 81)

= - 60 + 45

= - 15

Further,

n^th term, a_n = S_n - S_n-1

= [4n - n²] - [4 (n - 1) - (n - 1)² ]

= 4n - n² - 4n + 4 + (n - 1)²

= 4 - n² + n² - 2n + 1

= 5 - 2n

Therefore, a_n  = 5 - 2n