RD Sharma Class 11 Solutions Chapter - Some Special Series | Mathematics (Maths) Class 11 - Commerce PDF Download

 Page 1


21. Some Special Series
Exercise 21.1
1. Question
Find the sum of the following series to n terms:
1
3
 + 3
3
 + 5
3
 + 7
3
 + ……..
Answer
nth term would be = 2n - 1
We know, 
Therefore,
………….equation 1
…………….equation 2
From equation 1
=
[replace 2n by n]
=
Substituting in equation 2
2. Question
Find the sum of the following series to n terms:
2
3
 + 4
3
 + 6
3
 + 8
3
 + ………
Answer
Page 2


21. Some Special Series
Exercise 21.1
1. Question
Find the sum of the following series to n terms:
1
3
 + 3
3
 + 5
3
 + 7
3
 + ……..
Answer
nth term would be = 2n - 1
We know, 
Therefore,
………….equation 1
…………….equation 2
From equation 1
=
[replace 2n by n]
=
Substituting in equation 2
2. Question
Find the sum of the following series to n terms:
2
3
 + 4
3
 + 6
3
 + 8
3
 + ………
Answer
nth term would be 2n
We know ……(1)
Therefore,
Substituting the value from 1
3. Question
Find the sum of the following series to n terms:
1.2.5 + 2.3.6 + 3.4.7 + ……..
Answer
The nth term be n(n + 1)(n + 4)
Thus we can write 1.2.5 + 2.3.6 + 3.4.7 + ……..
The general term would be r(r + 1)(r + 4)
We know by property that:
?ax
n
 + bx
n - 1
 + cx
n - 2
…….d
0
=a?x
n
 + b?x
n - 1
 + c?x
n - 2
…….. + d
0
?1
Thus,
= ……(1)
We know
Substituting in (1)
Page 3


21. Some Special Series
Exercise 21.1
1. Question
Find the sum of the following series to n terms:
1
3
 + 3
3
 + 5
3
 + 7
3
 + ……..
Answer
nth term would be = 2n - 1
We know, 
Therefore,
………….equation 1
…………….equation 2
From equation 1
=
[replace 2n by n]
=
Substituting in equation 2
2. Question
Find the sum of the following series to n terms:
2
3
 + 4
3
 + 6
3
 + 8
3
 + ………
Answer
nth term would be 2n
We know ……(1)
Therefore,
Substituting the value from 1
3. Question
Find the sum of the following series to n terms:
1.2.5 + 2.3.6 + 3.4.7 + ……..
Answer
The nth term be n(n + 1)(n + 4)
Thus we can write 1.2.5 + 2.3.6 + 3.4.7 + ……..
The general term would be r(r + 1)(r + 4)
We know by property that:
?ax
n
 + bx
n - 1
 + cx
n - 2
…….d
0
=a?x
n
 + b?x
n - 1
 + c?x
n - 2
…….. + d
0
?1
Thus,
= ……(1)
We know
Substituting in (1)
4. Question
Find the sum of the following series to n terms:
1.2.4 + 2.3.7 + 3.4.10 + …………
Answer
The nth term be n(n + 1)(3n + 1)
1.2.4 + 2.3.7 + 3.4.10 + …………=
We know by property that:
?ax
n
 + bx
n - 1
 + cx
n - 2
…….d
0
=a?x
n
 + b?x
n - 1
 + c?x
n - 2
…….. + d
0
?1
 …..(1)
We know
Thus from (1)
5. Question
Find the sum of the following series to n terms:
1 + (1 + 2) + (1 + 2 + 3) + (1 + 2 + 3 + 4) + ………
Answer
The nth term be 
Where  = (n - 0) + (n - 1) + (n - 2) + …… + (n - n)
Page 4


21. Some Special Series
Exercise 21.1
1. Question
Find the sum of the following series to n terms:
1
3
 + 3
3
 + 5
3
 + 7
3
 + ……..
Answer
nth term would be = 2n - 1
We know, 
Therefore,
………….equation 1
…………….equation 2
From equation 1
=
[replace 2n by n]
=
Substituting in equation 2
2. Question
Find the sum of the following series to n terms:
2
3
 + 4
3
 + 6
3
 + 8
3
 + ………
Answer
nth term would be 2n
We know ……(1)
Therefore,
Substituting the value from 1
3. Question
Find the sum of the following series to n terms:
1.2.5 + 2.3.6 + 3.4.7 + ……..
Answer
The nth term be n(n + 1)(n + 4)
Thus we can write 1.2.5 + 2.3.6 + 3.4.7 + ……..
The general term would be r(r + 1)(r + 4)
We know by property that:
?ax
n
 + bx
n - 1
 + cx
n - 2
…….d
0
=a?x
n
 + b?x
n - 1
 + c?x
n - 2
…….. + d
0
?1
Thus,
= ……(1)
We know
Substituting in (1)
4. Question
Find the sum of the following series to n terms:
1.2.4 + 2.3.7 + 3.4.10 + …………
Answer
The nth term be n(n + 1)(3n + 1)
1.2.4 + 2.3.7 + 3.4.10 + …………=
We know by property that:
?ax
n
 + bx
n - 1
 + cx
n - 2
…….d
0
=a?x
n
 + b?x
n - 1
 + c?x
n - 2
…….. + d
0
?1
 …..(1)
We know
Thus from (1)
5. Question
Find the sum of the following series to n terms:
1 + (1 + 2) + (1 + 2 + 3) + (1 + 2 + 3 + 4) + ………
Answer
The nth term be 
Where  = (n - 0) + (n - 1) + (n - 2) + …… + (n - n)
Since,
 ………(1)
1 + (1 + 2) + (1 + 2 + 3) + (1 + 2 + 3 + 4) + ………….=
From (1)
1 + (1 + 2) + (1 + 2 + 3) + (1 + 2 + 3 + 4) + ………..
Thus, solving 
Solving 
We know by property that:
?ax
n
 + bx
n - 1
 + cx
n - 2
…….d
0
=a?x
n
 + b?x
n - 1
 + c?x
n - 2
…….. + d
0
?1
Thus,
We know,
Substituting
Page 5


21. Some Special Series
Exercise 21.1
1. Question
Find the sum of the following series to n terms:
1
3
 + 3
3
 + 5
3
 + 7
3
 + ……..
Answer
nth term would be = 2n - 1
We know, 
Therefore,
………….equation 1
…………….equation 2
From equation 1
=
[replace 2n by n]
=
Substituting in equation 2
2. Question
Find the sum of the following series to n terms:
2
3
 + 4
3
 + 6
3
 + 8
3
 + ………
Answer
nth term would be 2n
We know ……(1)
Therefore,
Substituting the value from 1
3. Question
Find the sum of the following series to n terms:
1.2.5 + 2.3.6 + 3.4.7 + ……..
Answer
The nth term be n(n + 1)(n + 4)
Thus we can write 1.2.5 + 2.3.6 + 3.4.7 + ……..
The general term would be r(r + 1)(r + 4)
We know by property that:
?ax
n
 + bx
n - 1
 + cx
n - 2
…….d
0
=a?x
n
 + b?x
n - 1
 + c?x
n - 2
…….. + d
0
?1
Thus,
= ……(1)
We know
Substituting in (1)
4. Question
Find the sum of the following series to n terms:
1.2.4 + 2.3.7 + 3.4.10 + …………
Answer
The nth term be n(n + 1)(3n + 1)
1.2.4 + 2.3.7 + 3.4.10 + …………=
We know by property that:
?ax
n
 + bx
n - 1
 + cx
n - 2
…….d
0
=a?x
n
 + b?x
n - 1
 + c?x
n - 2
…….. + d
0
?1
 …..(1)
We know
Thus from (1)
5. Question
Find the sum of the following series to n terms:
1 + (1 + 2) + (1 + 2 + 3) + (1 + 2 + 3 + 4) + ………
Answer
The nth term be 
Where  = (n - 0) + (n - 1) + (n - 2) + …… + (n - n)
Since,
 ………(1)
1 + (1 + 2) + (1 + 2 + 3) + (1 + 2 + 3 + 4) + ………….=
From (1)
1 + (1 + 2) + (1 + 2 + 3) + (1 + 2 + 3 + 4) + ………..
Thus, solving 
Solving 
We know by property that:
?ax
n
 + bx
n - 1
 + cx
n - 2
…….d
0
=a?x
n
 + b?x
n - 1
 + c?x
n - 2
…….. + d
0
?1
Thus,
We know,
Substituting
Thus the answer is 
6. Question
Find the sum of the following series to n terms:
1 × 2 + 2 × 3 + 3 × 4 + 4 × 5 + ………….
Answer
The last term be n(n + 1)
The generalized equation be
……………….(1)
Since We know by property that:
?ax
n
 + bx
n - 1
 + cx
n - 2
…….d
0
=a?x
n
 + b?x
n - 1
 + c?x
n - 2
…….. + d
0
?1
We know
Thus substituting in (1)
7. Question
Find the sum of the following series to n terms:
3 × 1
2
 + 5 × 2
2
 + 7 × 3
2
 + …………..
Answer
The nth term will be n
2
×(2n + 1)
The generalized equation be