NCERT Solutions For Class 11 Maths Chapter 4 Principle of Mathematical Induction Ex 4.1
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Textbook | NCERT |
Board | CBSE |
Category | NCERT Solutions |
Class | Class 10 |
Subject | Maths |
Chapter | Chapter 4 |
Exercise | Class 11 Maths Chapter 4 Principle of Mathematical Induction Ex 4.1 |
Number of Questions Solved | 24 |

NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction Ex 4.1
NCERT TEXTBOOK EXERCISES
Ex 4.1 Class 11 Maths Question 1. $1+{ 3 }^{ 2 }+{ 3 }^{ 3 }+\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +{ 3 }^{ n }=\frac { \left( { 3 }^{ n }-1 \right) }{ 2 }$
Solution.
Let the given statement be P(n) i.e.,
P(n) : $1+{ 3 }^{ 2 }+{ 3 }^{ 3 }+\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +{ 3 }^{ n }=\frac { \left( { 3 }^{ n }-1 \right) }{ 2 }$


Ex 4.1 Class 11 Maths Question 2. ${ 1 }^{ 3 }+{ 2 }^{ 3 }+{ 3 }^{ 3 }+\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +{ n }^{ 3 }={ \left( \frac { n\left( n+1 \right) }{ 2 } \right) }^{ 2 }$
Solution.
Let the given statement be P(n) i.e.,
P(n) : ${ 1 }^{ 3 }+{ 2 }^{ 3 }+{ 3 }^{ 3 }+\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +{ n }^{ 3 }={ \left( \frac { n\left( n+1 \right) }{ 2 } \right) }^{ 2 }$


Ex 4.1 Class 11 Maths Question 3. $1+\frac { 1 }{ \left( 1+2 \right) } +\frac { 1 }{ \left( 1+2+3 \right) } +\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +\frac { 1 }{ \left( 1+2+3+.\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +n \right) } =\frac { 2 }{ \left( n+1 \right) }$
Solution.
Let the given statement be P(n), i.e.,
P(n) : $1+\frac { 1 }{ \left( 1+2 \right) } +\frac { 1 }{ \left( 1+2+3 \right) } +\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +\frac { 1 }{ \left( 1+2+3+.\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +n \right) } =\frac { 2 }{ \left( n+1 \right) }$


Ex 4.1 Class 11 Maths Question 4. $1.2.3+2.3.4+\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +n\left( n+1 \right) \left( n+2 \right) =\frac { n\left( n+1 \right) \left( n+2 \right) \left( n+3 \right) }{ 4 }$
Solution.
Let the given statement be P(n), i.e.,
P(n) : $1.2.3+2.3.4+\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +n\left( n+1 \right) \left( n+2 \right) =\frac { n\left( n+1 \right) \left( n+2 \right) \left( n+3 \right) }{ 4 }$


Ex 4.1 Class 11 Maths Question 5. $1.3+{ 2.3 }^{ 2 }+{ 3.3 }^{ 3 }+\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +{ n.3 }^{ n }=\frac { \left( 2n-1 \right) { 3 }^{ n+1 }+3 }{ 4 }$
Solution.
Let the given statement be P(n), i.e.,
P(n) : $1.3+{ 2.3 }^{ 2 }+{ 3.3 }^{ 3 }+\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +{ n.3 }^{ n }=\frac { \left( 2n-1 \right) { 3 }^{ n+1 }+3 }{ 4 }$


Ex 4.1 Class 11 Maths Question 6. $1.2+2.3+3.4+\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +n.\left( n+1 \right) =\left[ \frac { n\left( n+1 \right) \left( n+2 \right) }{ 3 } \right]$
Solution.
Let the given statement be P(n), i.e.,
P(n) : $1.2+2.3+3.4+\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +n.\left( n+1 \right) =\left[ \frac { n\left( n+1 \right) \left( n+2 \right) }{ 3 } \right]$


Ex 4.1 Class 11 Maths Question 7. $1.3+3.5+5.7+\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +\left( 2n-1 \right) \left( 2n+1 \right) =\frac { n\left( { 4n }^{ 2 }+6n-1 \right) }{ 3 }$
Solution.
Let the given statement be P(n), i.e.,
P(n) : $1.3+3.5+5.7+\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +\left( 2n-1 \right) \left( 2n+1 \right) =\frac { n\left( { 4n }^{ 2 }+6n-1 \right) }{ 3 }$



Ex 4.1 Class 11 Maths Question 8. $1.2+2.{ 2 }^{ 2 }+3.{ 2 }^{ 3 }+\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +n.{ 2 }^{ n }=\left( n-1 \right) { 2 }^{ n+1 }+2$
Solution.
Let the given statement be P(n), i.e.,
P(n) : $1.2+2.{ 2 }^{ 2 }+3.{ 2 }^{ 3 }+\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +n.{ 2 }^{ n }=\left( n-1 \right) { 2 }^{ n+1 }+2$

Ex 4.1 Class 11 Maths Question 9 $\frac { 1 }{ 2 } +\frac { 1 }{ 4 } +\frac { 1 }{ 8 } +\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +\frac { 1 }{ { 2 }^{ n } } =1-\frac { 1 }{ { 2 }^{ n } }$
Solution.
Let the given statement be P(n), i.e.,
P(n) : $\frac { 1 }{ 2 } +\frac { 1 }{ 4 } +\frac { 1 }{ 8 } +\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +\frac { 1 }{ { 2 }^{ n } } =1-\frac { 1 }{ { 2 }^{ n } }$


Ex 4.1 Class 11 Maths Question 10. $\frac { 1 }{ 2.5 } +\frac { 1 }{ 5.8 } +\frac { 1 }{ 8.11 } +\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +\frac { 1 }{ \left( 3n-1 \right) \left( 3n+2 \right) } =\frac { n }{ \left( 6n+4 \right) }$
Solution.
Let the given statement be P(n), i.e.,
P(n) : $\frac { 1 }{ 2.5 } +\frac { 1 }{ 5.8 } +\frac { 1 }{ 8.11 } +\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +\frac { 1 }{ \left( 3n-1 \right) \left( 3n+2 \right) } =\frac { n }{ \left( 6n+4 \right) }$


Ex 4.1 Class 11 Maths Question 11. $\frac { 1 }{ 1.2.3 } +\frac { 1 }{ 2.3.4 } +\frac { 1 }{ 3.4.5 } +\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +\frac { 1 }{ n\left( n+1 \right) \left( n+2 \right) } =\frac { n\left( n+3 \right) }{ 4\left( n+1 \right) \left( n+2 \right) }$
Solution.
Let the given statement be P(n), i.e.,
P(n) : $\frac { 1 }{ 1.2.3 } +\frac { 1 }{ 2.3.4 } +\frac { 1 }{ 3.4.5 } +\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +\frac { 1 }{ n\left( n+1 \right) \left( n+2 \right) } =\frac { n\left( n+3 \right) }{ 4\left( n+1 \right) \left( n+2 \right) }$



Ex 4.1 Class 11 Maths Question 12. $a+ar+{ ar }^{ 2 }+\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +{ ar }^{ n-1 }=\frac { a\left( { r }^{ n }-1 \right) }{ r-1 }$
Solution.
Let the given statement be P(n), i.e.,
P(n) : $a+ar+{ ar }^{ 2 }+\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +{ ar }^{ n-1 }=\frac { a\left( { r }^{ n }-1 \right) }{ r-1 }$


Ex 4.1 Class 11 Maths Question 13. $\left( 1+\frac { 3 }{ 1 } \right) \left( 1+\frac { 5 }{ 4 } \right) \left( 1+\frac { 7 }{ 9 } \right) \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \left( 1+\frac { \left( 2n+1 \right) }{ { n }^{ 2 } } \right) ={ \left( n+1 \right) }^{ 2 }$
Solution.
Let the given statement be P(n), i.e.,
P(n) : $\left( 1+\frac { 3 }{ 1 } \right) \left( 1+\frac { 5 }{ 4 } \right) \left( 1+\frac { 7 }{ 9 } \right) \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \left( 1+\frac { \left( 2n+1 \right) }{ { n }^{ 2 } } \right) ={ \left( n+1 \right) }^{ 2 }$


Ex 4.1 Class 11 Maths Question 14. $\left( 1+\frac { 1 }{ 1 } \right) \left( 1+\frac { 1 }{ 2 } \right) \left( 1+\frac { 1 }{ 3 } \right) \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \left( 1+\frac { 1 }{ n } \right) =\left( n+1 \right)$
Solution.
Let the given statement be P(n), i.e.,
P(n) : $\left( 1+\frac { 1 }{ 1 } \right) \left( 1+\frac { 1 }{ 2 } \right) \left( 1+\frac { 1 }{ 3 } \right) \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \left( 1+\frac { 1 }{ n } \right) =\left( n+1 \right)$


Ex 4.1 Class 11 Maths Question 15. ${ 1 }^{ 2 }+{ 3 }^{ 2 }+{ 5 }^{ 2 }+\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +{ \left( 2n-1 \right) }^{ 2 }=\frac { n\left( 2n-1 \right) \left( 2n+1 \right) }{ 3 }$
Solution.
Let the given statement be P(n), i.e.,
P(n) : ${ 1 }^{ 2 }+{ 3 }^{ 2 }+{ 5 }^{ 2 }+\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +{ \left( 2n-1 \right) }^{ 2 }=\frac { n\left( 2n-1 \right) \left( 2n+1 \right) }{ 3 }$


Ex 4.1 Class 11 Maths Question 16. $\frac { 1 }{ 1.4 } +\frac { 1 }{ 4.7 } +\frac { 1 }{ 7.10 } +\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +\frac { 1 }{ \left( 3n-2 \right) \left( 3n+1 \right) } =\frac { n }{ \left( 3n+1 \right) }$
Solution.
Let the given statement be P(n), i.e.,
P(n) : $\frac { 1 }{ 1.4 } +\frac { 1 }{ 4.7 } +\frac { 1 }{ 7.10 } +\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +\frac { 1 }{ \left( 3n-2 \right) \left( 3n+1 \right) } =\frac { n }{ \left( 3n+1 \right) }$


Ex 4.1 Class 11 Maths Question 17. $\frac { 1 }{ 3.5 } +\frac { 1 }{ 5.7 } +\frac { 1 }{ 7.9 } +\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +\frac { 1 }{ \left( 2n+1 \right) \left( 2n+3 \right) } =\frac { n }{ 3\left( 2n+3 \right) }$
Solution.
Let the given statement be P(n), i.e.,
P(n) : $\frac { 1 }{ 3.5 } +\frac { 1 }{ 5.7 } +\frac { 1 }{ 7.9 } +\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot +\frac { 1 }{ \left( 2n+1 \right) \left( 2n+3 \right) } =\frac { n }{ 3\left( 2n+3 \right) }$


Ex 4.1 Class 11 Maths Question 18. 1+2+3+⋅⋅⋅⋅⋅⋅⋅⋅+n<1/8(2n+1)2
Solution.
Let the given statement be P(n), i.e.,
P(n) : 1+2+3+⋅⋅⋅⋅⋅⋅⋅⋅+n<1/8(2n+1)2


Ex 4.1 Class 11 Maths Question 19. n(n+1 )(n + 5) is a multiple of 3.
Solution.
Let the given statement be P(n), i.e.,
P(n): n(n + l)(n + 5) is a multiple of 3.


Ex 4.1 Class 11 Maths Question 20. 102n−1+1 is divisible by 11.
Solution.
Let the given statement be P(n), i.e.,
P(n): 102n−1+1 is divisible by 11

Ex 4.1 Class 11 Maths Question 21. x2n−y2n is divisible by x + y.
Solution.
Let the given statement be P(n), i.e.,
P(n): x2n−y2n is divisible by x + y.

Ex 4.1 Class 11 Maths Question 22. 32n+2−8n−9 is divisible by 8.
Solution.
Let the given statement be P(n), i.e.,
P(n): 32n+2−8n−9 is divisible by 8.

Ex 4.1 Class 11 Maths Question 23. 41n−14n is a multiple of 27.
Solution.
Let the given statement be P(n), i.e.,
P(n): 41n−14n is a multiple of 27.

Ex 4.1 Class 11 Maths Question 24. (2n+7)<(n+3)2
Solution.
Let the given statement be P(n), i.e.,
P(n): (2n+7)<(n+3)2
First we prove that the statement is true for n = 1.

NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction Exercise 4.1 PDF
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