Chapter 9 – Sequences and Series (Ex – 9.2)

Sequences and Series (Ex – 9.2)

Question 1.
Find the sum of odd integers from 1 to 2001.

Solution:
We have to find 1 + 3 + 5 + ……….. + 2001
This is an A.P. with first term a = 1, common difference d = 3-1 = 2 and last term l = 2001
∴ l = a + (n-1 )d
⇒ 2001 = 1 + (n -1)2
⇒ 2001 = 1 + 2n – 2
⇒ 2n = 2001 + 1

Question 2.
Find the sum of all natural numbers lying between 100 and 1000, which are multiples of 5.

Solution:
We have to find 105 +110 +115 + ……..+ 995
This is an A.P. with first term a = 105, common difference d = 110 -105 = 5 and last term 1 = 995

Question 3.
In an A.P., the first term is 2 and the sum of the first five terms is one-fourth of the next five terms. Show that 20th term is -112.

Solution:
Let a = 2 be the first term and d be the common difference.

Question 4.
How many terms of the A.P. -6, −11/2, -5, ……. are needed to give the sum – 25 ?

Solution:
Let a be the first term and d be the common difference of the given A.P., we have

Question 5.
In an A.P., if pth term is 1/q and qth term is 1/p prove that the sum of first pq terms is 1/2(pq+1), where p±q.

Solution:
Let a be the first term & d be the common difference of the A.P., then

Question 6.
If the sum of a certain number of terms of the A.P. 25, 22, 19, …. is 116. Find the last term.

Solution:
Let a be the first term and d be the common difference.
We have a = 25, d = 22 – 25 = -3, S_n = 116

Question 7.
Find the sum to n terms of the A.P., whose k^th term is 5k + 1.

Solution:
We have a_k = 5k +1
By substituting the value of k = 1, 2, 3 and 4,
we get

Question 8.
If the sum of n terms of an A.P. is (pn + qn²), where p and q are constants, find the common difference.

Solution:
We have S_n = pn + qn², where S„ be the sum of n terms.

Question 9.
The sums of n terms of two arithmetic progressions are in the ratio 5n + 4 : 9n + 6. Find the ratio of their 18^th terms.

Solution:
Let a₁, a₂ & d₁ d₂ be the first terms & common differences of the two arithmetic progressions respectively. According to the given condition, we have

Question 10.
If the sum of first p terms of an A.P. is equal to the sum of the first q terms, then find the sum of the first (p + q) terms.

Solution:
Let the first term be a and common difference be d.
According to question

Question 11.
Sum of the first p, q and r terms of an A.P. are a, b and c, respectively.
Prove that

Solution.
Let the first term be A & common difference be D. We have

Question 12.
The ratio of the sums of m and n terms of an A.P. is m²: n². Show that the ratio of m^th and n^th term is (2m -1): (2n -1).

Solution:
Let the first term be a & common difference be d. Then

Question 13.
If the sum of n terms of an A.P. is 3n² + 5n and its mth term is 164, find the value of m.

Solution:
We have S_n = 3n² + 5n, where Sn be the sum of n terms.

Question 14.
Insert five numbers between 8 and 26 such that the resulting sequence is an A.P.

Solution:
Let A₁, A₂, A₃, A₄, A₅ be numbers between 8 and 26 such that 8, A₁, A₂, A₃, A₄, A₅, 26 are in A.P., Here a = 8, l = 26, n = 7

Question 15.

Solution:

Question 16.
Between 1 and 31, m numbers have been inserted in such a way that the resulting sequence is an A.P. and the ratio of 7^thand (m – 1 )^th numbers is 5:9. Find the value of m.

Solution:
Let the sequence be 1, A₁, A₂, ……… A_m, 31 Then 31 is (m + 2)^th term, a = 1, let d be the common difference

Question 17.
A man starts repaying a loan as first instalment of Rs. 100. If he increases the instalment by Rs. 5 every month, what amount he will pay in the 30th instalment?

Solution:
Here, we have an A.P. with a = 100 and d = 5
∴ a_n= a + 29d = 100 + 29(5) = 100 + 145 = 245
Hence he will pay Rs. 245 in 30^th instalment.

Question 18.
The difference between any two consecutive interior angles of a polygon is 5°. If the smallest angle is 120°, find the number of the sides of the polygon.

Solution:
Let there are n sides of a polygon.