Question

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

If S_n = 4n – n², then 

(i) S₁ = a =? 

(ii) S₂ =? 

(iii) a₂ =? 

(iv) a₃ =? 

(v) a₁₀ =? 

(vi) a_n =?

Answer 1

(i) S_n = 4n – n² 

S₁ = 4(1) – 12 = 4 – 1 

S₁ = 3 

∴ S₁ = a = 3. 

(ii) S₂ = 4n – n² 

S₂ = 4(2) – 22 = 8 – 4 

S2 = 4 

∴ S₂ = 4 

(iii) We have 

S₂ = a₁ + a₂ = 4 

= 3 + a₂ = 4 

∴ a₂ = 4 – 3 

∴ a₂ = 1 

(iv) S_n = 4n – n² 

S₃ = 4(3) – 3² 

= 12 – 9 

S₃ = 3 

a₁ + a₂ + a₃ = 3 

3 + 1 + a₃ = 3 

4 + a₃= 3 

∴ a₃ = 3 – 4 

∴ a₃ = -1 

(v) d = a₃ – a₃ = -1 -1 = -2 

a₁₀= a + 9d 

= 3 + 9(-2) 

= 3 – 18 

a₁₀ = -15 

∴ a₁₀ = -15 

(vi) a_n = a + (n – 1)d 

= 3 + (n – 1)(-2) 

= 3 + 2n + 2 

∴ a_n = 5 – 2n 

∴ a_n = 5 – 2n