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

Solution:

Given, S_n = 4n – n²

Therefore, S₁ = 4(1) – (1)² = 4 – 1 = 3 (Answer)

And S₂ = 4(2) – (2)² = 8 – 4 = 4

∴ Second term = a₂

= S₂ – S₁ [∵ a_n = S_n – S_{n – 1}]

= 4 – 3

= 1 (Answer)

Similarly, S₃ = 4(3) – (3)² = 12 – 9 = 3(Answer)

S₉ = 4(9) – (9)² = 36 – 81 = -45

S₁₀ = 4(10) – (10)² = 40 – 100 = -60

Therefore,

Third term = a₃ = S₃ – S₂ [∵ a_n = S_n – S_{n – 1}]

= 3 – 4

= -1 (Answer)

Tenth term = a₁₀ = S₁₀ – S₉ [∵ a_n = S_n – S_{n – 1}]

= -60 – (-45)

= -60 + 45

= -15 (Answer)

nth term = a_n = S_n – S_{n – 1}

= 4n – n² – {4(n – 1) – (n – 1)²}

= 4n – n² – {4n – 4 – (n² – 2n + 1)}

= 4n – n² – {4n – 4 – n² + 2n – 1}

= 4n – n² – (- n² + 6n – 5)

= 4n – n² + n² – 6n + 5

= 5 – 2n (Answer)