Question

Q27) If the sum of the first n terms of an AP is on - n, what is the first term? What is the sum of first two
terms? Similarly, find the 3rd, the 10th and the nth terms.
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Answer 1

$\huge{\mathcal{\underline{\green{ANSWER}}}}$

For any AP, sum upto first n terms,

Sn= (n/2)[2a+(n-1)d] ——(1)

For given AP, sum of the first n terms, Sn= 4n-n²

First term, a = a₁ =S₁= 4(1)-(1)² = 4–1=3

Sum of first 2 terms, S₂ = 4(2)-(2)² = 8–4=4

Second term, a₂= S₂-S₁ = 4–3 =1

Sum of first 3 terms, S₃ = 4(3)-(3)² =12–9=3

3rd term, a₃=S₃-S₂ = 3–4=-1

Therefore, the series is: 3, 1, -1

Common difference, d =a₃-a₂= a₂-a₁= -2

S₉=4(9)-(9)²= 36–81 = -45

S₁₀=4(10)-(10)²= 40–100 = -60

Alternatively, using eqn (1), with a = 3 and d= -2

S₉= (9/2)[2(3)+(9-1)(-2)]= -45

S₁₀= (10/2)[2(3)+(10–1)(-2)]= -60

Now, 10 th term, a₁₀=S₁₀-S₉ = -60–(-45)=-15

nth term, aₙ = a+(n-1) d= 3+(n-1)(-2)= 5–2n

Ans: 3rd term =-1; 10th term = -15 ; nth term = (5–2n)