Question

If the sum of the first n terms of an AP is 4n – n² , what is the first term (remember the first term is S₁)? 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.

Answer 1

Given an A.P in which S_n = 4n – n² 

Taking n = 1 we get 

S₁ = 4 × 1- 12 = 4 – 1 = 3 

n = 2; S₂ = a₁ + a₂ = 4 × 2 – 2² = 8 – 4 = 4 

n = 3; S₃ = a₁ + a₂ + a₃ = 4 × 3 -3² = 12 – 9 = 3 

n = 4; S₄ = a₁ + a₂ + a₃ + a₄ = 4 × 4 – 4² = 16 – 16 = 0

Hence, S₁ = a₁ = 3 

a₂ = S₂ – S₁ = 4 – 3 = 1 

a₃ = S₃ – S₂ = 3 – 4 = -1 

a₄ = S₄ – S₃ = 0 – 3 = -3 

So, d = a₂ – a₁ = l – 3 = -2 

Now, a₁₀ = a + 9d  [∵ a_n = a + (n – 1) d] 

= 3 + 9 × (- 2) =

 3 – 18 = -15 

a_n = 3 + (n – 1) × (-2)

= 3 – 2n + 2 

= 5 – 2n