NCERT Solutions Class 11th Maths Chapter – 9 Sequences and Series Exercise 9.2

NCERT Solutions Class 11th Maths Chapter – 9 Sequences and Series Exercise 9.2

NCERT Solutions Class 11th Maths Chapter – 9 Sequences and Series Exercise 9.2

?Chapter – 9?

✍Sequences and Series✍

?Exercise 9.2?

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

?‍♂️solution – The odd integers from 1 to 2001 are 1, 3, 5, …1999, 2001.
It clearly forms a sequence in A.P.
Where, the first term, a = 1
Common difference, d = 2
Now,
a + (n -1)d = 2001
1 + (n-1)(2) = 2001
2n – 2 = 2000
2n = 2000 + 2 = 2002
n = 1001
We know,
S_n = n/2 [2a + (n-1)d]

Therefore, the sum of odd numbers from 1 to 2001 is 1002001.

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

?‍♂️solution – The natural numbers lying between 100 and 1000, which are multiples of 5, are 105, 110, … 995.
It clearly forms a sequence in A.P.
Where, the first term, a = 105
Common difference, d = 5
Now,
a + (n -1)d = 995
105 + (n – 1)(5) = 995
105 + 5n – 5 = 995
5n = 995 – 105 + 5 = 895
n = 895/5
n = 179
We know,
S_n = n/2 [2a + (n-1)d]

Therefore, the sum of all natural numbers lying between 100 and 1000, which are multiples of 5, is 98450.

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 20^th term is –112.

?‍♂️solution – Given,
The first term (a) of an A.P = 2
Let’s assume d be the common difference of the A.P.
So, the A.P. will be 2, 2 + d, 2 + 2d, 2 + 3d, …
Then,
Sum of first five terms = 10 + 10d
Sum of next five terms = 10 + 35d
From the question, we have
10 + 10d = ¼ (10 + 35d)
40 + 40d = 10 + 35d
30 = -5d
d = -6
a₂₀ = a + (20 – 1)d = 2 + (19) (-6) = 2 – 114 = -112
Therefore, the 20^th term of the A.P. is –112.

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

?‍♂️solution – Let’s consider the sum of n terms of the given A.P. as –25.
We known that,
S_n = n/2 [2a + (n-1)d]
where n = number of terms, a = first term, and d = common difference
So here, a = –6
d = -11/2 + 6 = (-11 + 12)/ 2 = 1/2
Thus, we have

5. In an A.P., if p^th term is 1/q and q^th term is 1/p, prove that the sum of first pq terms is ½ (pq + 1) where p ≠ q.  

?‍♂️solution – 

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

?‍♂️solution – Given A.P.,
25, 22, 19, …
Here,
First term, a = 25 and
Common difference, d = 22 – 25 = -3
Also given, sum of certain number of terms of the A.P. is 116
The number of terms be n
So, we have
S_n = n/2 [2a + (n-1)d] = 116
116 = n/2 [2(25) + (n-1)(-3)]
116 x 2 = n [50 – 3n + 3]
232 = n [53 – 3n]
232 = 53n – 3n²
3n² – 53n + 232 = 0
3n² – 24n – 29n+ 232 = 0
3n(n – 8) – 29(n – 8) = 0
(3n – 29) (n – 8) = 0
Hence,
n = 29/3 or n = 8
As n can only be an integral value, n = 8
Thus, 8th term is the last term of the A.P.
a8 = 25 + (8 – 1)(-3)
= 25 – 21
= 4

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

?‍♂️solution – Given, the k^th term of the A.P. is  5k + 1.
k^th term = a_k = a + (k – 1)d
And,
a + (k – 1)d =  5k + 1
a + kd – d =  5k + 1
On comparing the coefficient of k, we get d = 5
a – d = 1
a – 5 = 1
⇒ a = 6

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 know that,
S_n = n/2 [2a + (n-1)d]
From the question we have,

On comparing the coefficients of n² on both sides, we get
d/2 = q
Hence, d = 2q
Therefore, the common difference of the A.P. is 2q.

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 – Leta₁, a₂, and d₁, d₂ be the first terms and the common difference of the first and second arithmetic progression respectively.
Then, from the question we have

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’s take a and d to be the first term and the common difference of the A.P. respectively.
Then, it given that

Therefore, the sum of (p + q) terms of the A.P. is 0.

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

Prove that 

?‍♂️solution – Let a₁ and d be the first term and the common difference of the A.P. respectively.
Then according to the question, we have

Now, subtracting (2) from (1), we get

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’s consider that a and b to be the first term and the common difference of the A.P. respectively.
Then from the question, we have

Hence, the given result is proved.

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

?‍♂️solution – Let’s consider a and b to be the first term and the common difference of the A.P. respectively.
a_m = a + (m – 1)d = 164 … (1)
We the sum of the terms is given by,
S_n = n/2 [2a + (n-1)d]

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

?‍♂️solution – Let’s assume A₁, A₂, A₃, A₄, and A₅ to be five numbers between 8 and 26 such that 8, A₁, A₂, A₃, A₄, A₅, 26 are in an A.P.
Here we have,
a = 8, b = 26, n = 7
So,
26 = 8 + (7 – 1) d
6d = 26 – 8 = 18
d = 3
Now,
A₁ = a + d = 8 + 3 = 11
A₂ = a + 2d = 8 + 2 × 3 = 8 + 6 = 14
A₃ = a + 3d = 8 + 3 × 3 = 8 + 9 = 17
A₄ = a + 4d = 8 + 4 × 3 = 8 + 12 = 20
A₅ = a + 5d = 8 + 5 × 3 = 8 + 15 = 23
Therefore, the required five numbers between 8 and 26 are 11, 14, 17, 20, and 23.

15. If is the A.M. between a and b, then find the value of n.

?‍♂️solution – The A.M between a and b is given by, (a + b)/2
Then according to the question,

Thus, the value of n is 1.

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^th and (m – 1)^th numbers is 5: 9. Find the value of m.

?‍♂️solution – Let’s consider a₁, a₂, … a_m be m numbers such that 1, a₁, a₂, … a_m, 31 is an A.P.
And here,
a = 1, b = 31, n = m + 2
So, 31 = 1 + (m + 2 – 1) (d)
30 = (m + 1) d
d = 30/ (m + 1) ……. (1)
Now,
a₁ = a + d
a₃ = a + 3d …
a₃ = a + 3d …
Hence, a₇ = a + 7d
a_m–1 = a + (m – 1) d
According to the question, we have

Therefore, the value of m is 14.

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 30^th instalment?

?‍♂️solution – Given,
The first instalment of the loan is Rs 100.
The second instalment of the loan is Rs 105 and so on as the instalment increases by Rs5 every month.
Thus, the amount that the man repays every month forms an A.P.
And the, A.P. is 100, 105, 110, …
Where, first term, a = 100
Common difference, d = 5
So, the 30th term in this A.P. will be
A₃₀  = a + (30 – 1)d
= 100 + (29) (5)
= 100 + 145
= 245
Therefore, the amount to be paid in the 30^th  instalment will be Rs 245.

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 – It’s understood from the question that, the angles of the polygon will form an A.P. with common difference d = 5° and first term a = 120°.
And, we know that the sum of all angles of a polygon with n sides is 180° (n – 2).
Thus, we can say

Thus, a polygon having 9 and 16 sides will satisfy the condition in the question.