NCERT MATHS CLASS - X
Arithmetic Progressions
Exercise 5.3
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 3rd, the10th and the nth terms.
Sol.
Sol.
Given, Sn = 4n − n2
Let first term be a,
Therefore, S1 =a = 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
Therefore, 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.
Question 12:
Find the sum of first 40 positive integers divisible by 6.
Sol.
Sol.
The positive integers, divisible by 6 are 6, 12, 18, 24 …
The above series is an A.P. whose first term is 6 and common difference is 6.
a = 6, d = 6, S40 =?
= 20[12 + (39) (6)]
= 20(12 + 234)
= 20 × 246
= 4920
Question 13 :
Find the sum of first 15 multiples of 8.
Sol.
Sol.
The multiples of 8 are 8, 16, 24, 32…
The series is in A.P with first term 8 and common difference 8.
Therefore, a = 8, d = 8, S15 =?
= 960
Question 14:
Find the sum of the odd numbers between 0 and 50.
Sol.
Sol.
The odd numbers between 0 and 50 are
1, 3, 5, 7, 9 … 49, the series is in A.P.
a = 1, d = 2, l = 49
l = a + (n − 1) d
49 = 1 + (n − 1)2
48 = 2(n − 1)
n − 1 = 24
n = 25
