Question

Question 11 If the sum of the first n terms of an AP is 4n − n^2 , 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.

Class 10 - Math - Arithmetic Progressions Page 113

Answer 1

Given that,
Sn = 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
an = 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. 3rd, 10th, and nth terms are −1, −15, and 5 − 2n respectively

Answer 2

Hi !

Given ,

Sn = 4n − n²

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

Sum of first two terms , a₁ + a₂ = S₂

= 4(2) − (2)² = 8 − 4 = 4

Second term, a₂ = S₂ − S₁
= 4 − 3
= 1
=============================
d = a₂ - a₁ = 1 - 3 = -2
d = -2

an = a + (n − 1)d
= 3 + (n − 1) (−2)
= 3 − 2n + 2
= 5 − 2n

Therefore,
an = 5 - 2n

[ n = 3 ]
a₃ = 5 − 2(3) = 5 − 6 = −1

[ n = 10 ]
a₁₀ = 5 − 2(10) = 5 − 20 = −15
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first term = a = 3
sum of first two terms = S₂ = 4

3rd term = a₃ = -1
10th term = -15

n^th term = 5 - 2n