Question 11:

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, the10^th and the n^th terms.

Answer:

Given that,

S_n = 4n − n²

First term, a = S₁ = 4(1) − (1)² = 4 − 1 = 3

Sum of first two terms = S2

= 4(2) − (2)² = 8 − 4 = 4

Second term, a₂ = S₂ − S₁ = 4 − 3 = 1

d = a₂ − a = 1 − 3 = −2

a_n = a + (n − 1)d

= 3 + (n − 1) (−2)

= 3 − 2n + 2

= 5 − 2n

Therefore, a₃ = 5 − 2(3) = 5 − 6 = −1

a₁₀ = 5 − 2(10) = 5 − 20 = −15

Hence, the sum of first two terms is 4. The second term is 1. 3^rd, 10^th, and n^th terms are −1, −15, and 5 − 2n respectively.

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