NCERT Textbook: Principle of Mathematical Induction | NCERT Textbooks (Class 6 to Class 12) - CTET & State TET PDF Download

 Page 1


vAnalysis and natural philosophy owe their most important discoveries to
this fruitful means, which is called induction. Newton was indebted
to it for his theorem of the binomial and the principle of
universal gravity. – LAPLACE v
4.1  Introduction
One key basis for mathematical thinking is deductive rea-
soning. An informal, and example of deductive reasoning,
borrowed from the study of logic, is an argument expressed
in three statements:
(a) Socrates is a man.
(b) All men are mortal, therefore,
(c) Socrates is mortal.
If statements (a) and (b) are true, then the truth of (c) is
established. To make this simple mathematical example,
we could write:
(i) Eight is divisible by two.
(ii) Any number divisible by two is an even number,
therefore,
(iii) Eight is an even number.
Thus, deduction in a nutshell is given a statement to be proven, often called a
conjecture or a theorem in mathematics, valid deductive steps are derived and a
proof may or may not be established, i.e., deduction is the application of a general
case to a particular case.
In contrast to deduction, inductive reasoning depends on working with each case,
and developing a conjecture by observing incidences till we have observed each and
every case. It is frequently used in mathematics and is a key aspect of scientific
reasoning, where collecting and analysing data is the norm. Thus, in simple language,
we can say the word induction means the generalisation from particular cases or facts.
Chapter
4
PRINCIPLE OF
MATHEMATICAL INDUCTION
G . Peano
(1858-1932)
2022-23
Page 2


vAnalysis and natural philosophy owe their most important discoveries to
this fruitful means, which is called induction. Newton was indebted
to it for his theorem of the binomial and the principle of
universal gravity. – LAPLACE v
4.1  Introduction
One key basis for mathematical thinking is deductive rea-
soning. An informal, and example of deductive reasoning,
borrowed from the study of logic, is an argument expressed
in three statements:
(a) Socrates is a man.
(b) All men are mortal, therefore,
(c) Socrates is mortal.
If statements (a) and (b) are true, then the truth of (c) is
established. To make this simple mathematical example,
we could write:
(i) Eight is divisible by two.
(ii) Any number divisible by two is an even number,
therefore,
(iii) Eight is an even number.
Thus, deduction in a nutshell is given a statement to be proven, often called a
conjecture or a theorem in mathematics, valid deductive steps are derived and a
proof may or may not be established, i.e., deduction is the application of a general
case to a particular case.
In contrast to deduction, inductive reasoning depends on working with each case,
and developing a conjecture by observing incidences till we have observed each and
every case. It is frequently used in mathematics and is a key aspect of scientific
reasoning, where collecting and analysing data is the norm. Thus, in simple language,
we can say the word induction means the generalisation from particular cases or facts.
Chapter
4
PRINCIPLE OF
MATHEMATICAL INDUCTION
G . Peano
(1858-1932)
2022-23
PRINCIPLE OF MATHEMATICAL INDUCTION       87
In algebra or in other discipline of mathematics, there are certain results or state-
ments that are formulated in terms of n, where n is a positive integer. To prove such
statements the well-suited principle that is used–based on the specific technique, is
known as the principle of mathematical induction.
4.2  Motivation
In mathematics, we use a form of complete induction called mathematical induction.
To understand the basic principles of mathematical induction, suppose a set of thin
rectangular tiles are placed as shown in Fig 4.1.
Fig 4.1
When the first tile is pushed in the indicated direction, all the tiles will fall. To be
absolutely sure that all the tiles will fall, it is sufficient to know that
(a) The first tile falls, and
(b) In the event that any tile falls its successor necessarily falls.
This is the underlying principle of mathematical induction.
We know, the set of natural numbers N is a special ordered subset of the real
numbers. In fact, N is the smallest subset of R with the following property:
A set S is said to be an inductive set if 1? S and  x + 1 ? S whenever x ? S. Since
N is the smallest subset of R which is an inductive set, it follows that any subset of R
that is an inductive set must contain N.
Illustration
Suppose we wish to find the formula for the sum of positive integers 1, 2, 3,...,n, that is,
a formula which will give the value of 1 + 2 + 3 when n = 3, the value 1 + 2 + 3 + 4,
when n = 4 and so on and suppose that in some manner we are led to believe that the
formula 1 + 2 + 3+...+ n = 
( 1)
2
n n +
 is the correct one.
How can this formula actually be proved? We can, of course, verify the statement
for as many positive integral values of n as we like, but this process will not prove the
formula for all values of n. What is needed is some kind of chain reaction which will
2022-23
Page 3


vAnalysis and natural philosophy owe their most important discoveries to
this fruitful means, which is called induction. Newton was indebted
to it for his theorem of the binomial and the principle of
universal gravity. – LAPLACE v
4.1  Introduction
One key basis for mathematical thinking is deductive rea-
soning. An informal, and example of deductive reasoning,
borrowed from the study of logic, is an argument expressed
in three statements:
(a) Socrates is a man.
(b) All men are mortal, therefore,
(c) Socrates is mortal.
If statements (a) and (b) are true, then the truth of (c) is
established. To make this simple mathematical example,
we could write:
(i) Eight is divisible by two.
(ii) Any number divisible by two is an even number,
therefore,
(iii) Eight is an even number.
Thus, deduction in a nutshell is given a statement to be proven, often called a
conjecture or a theorem in mathematics, valid deductive steps are derived and a
proof may or may not be established, i.e., deduction is the application of a general
case to a particular case.
In contrast to deduction, inductive reasoning depends on working with each case,
and developing a conjecture by observing incidences till we have observed each and
every case. It is frequently used in mathematics and is a key aspect of scientific
reasoning, where collecting and analysing data is the norm. Thus, in simple language,
we can say the word induction means the generalisation from particular cases or facts.
Chapter
4
PRINCIPLE OF
MATHEMATICAL INDUCTION
G . Peano
(1858-1932)
2022-23
PRINCIPLE OF MATHEMATICAL INDUCTION       87
In algebra or in other discipline of mathematics, there are certain results or state-
ments that are formulated in terms of n, where n is a positive integer. To prove such
statements the well-suited principle that is used–based on the specific technique, is
known as the principle of mathematical induction.
4.2  Motivation
In mathematics, we use a form of complete induction called mathematical induction.
To understand the basic principles of mathematical induction, suppose a set of thin
rectangular tiles are placed as shown in Fig 4.1.
Fig 4.1
When the first tile is pushed in the indicated direction, all the tiles will fall. To be
absolutely sure that all the tiles will fall, it is sufficient to know that
(a) The first tile falls, and
(b) In the event that any tile falls its successor necessarily falls.
This is the underlying principle of mathematical induction.
We know, the set of natural numbers N is a special ordered subset of the real
numbers. In fact, N is the smallest subset of R with the following property:
A set S is said to be an inductive set if 1? S and  x + 1 ? S whenever x ? S. Since
N is the smallest subset of R which is an inductive set, it follows that any subset of R
that is an inductive set must contain N.
Illustration
Suppose we wish to find the formula for the sum of positive integers 1, 2, 3,...,n, that is,
a formula which will give the value of 1 + 2 + 3 when n = 3, the value 1 + 2 + 3 + 4,
when n = 4 and so on and suppose that in some manner we are led to believe that the
formula 1 + 2 + 3+...+ n = 
( 1)
2
n n +
 is the correct one.
How can this formula actually be proved? We can, of course, verify the statement
for as many positive integral values of n as we like, but this process will not prove the
formula for all values of n. What is needed is some kind of chain reaction which will
2022-23
88       MATHEMATICS
have the effect that once the formula is proved for a particular positive integer the
formula will automatically follow for the next positive integer and the next  indefinitely.
Such a reaction may be considered as produced by the method of mathematical induction.
4.3  The Principle of Mathematical Induction
Suppose there is a given statement P(n)  involving the natural number n such that
(i) The statement is true for n = 1, i.e., P(1) is true, and
(ii) If the statement is true for n = k (where k is some positive integer), then
the statement is also true for n = k + 1, i.e., truth of P(k) implies the
truth of P (k + 1).
Then, P(n) is true for all natural numbers n.
Property (i) is simply  a statement of fact. There may be situations when a
statement is true for all n = 4. In this case, step 1 will start from n = 4 and we shall
verify the result for n = 4, i.e., P(4).
Property (ii) is a conditional property. It does not assert that the given statement
is true for n = k, but only that if it is true for n = k, then it is also true for n = k +1. So,
to prove that the  property holds , only prove that conditional proposition:
If the statement is true for n = k, then it is also true for n = k + 1.
This is sometimes referred to as the inductive step. The assumption that the given
statement is true for n = k in this inductive step is called the inductive hypothesis.
For example, frequently in mathematics, a formula will be discovered that appears
to fit a pattern like
1 = 1
2   
=1
4 = 2
2
 = 1 + 3
9 = 3
2
 = 1 + 3 + 5
16 = 4
2
 = 1 + 3 + 5 + 7, etc.
It is worth to be noted that the sum of the first two odd natural numbers is the
square of second natural number, sum of the first three odd natural numbers is the
square of third natural number and so on.Thus, from this pattern it appears that
1 + 3 + 5 + 7 + ... + (2n – 1) = n
2 
, i.e,
the sum of the first n odd natural numbers is the square of n.
Let us write
P(n): 1 + 3 + 5 + 7 + ... + (2n – 1) = n
2
.
We wish to prove that P(n) is true for all n.
The first step in a proof that uses mathematical induction is to prove that
P (1) is true. This step is called the basic step. Obviously
1 = 1
2
, i.e., P(1) is true.
The next step is called the inductive step. Here, we suppose that P (k) is true for some
2022-23
Page 4


vAnalysis and natural philosophy owe their most important discoveries to
this fruitful means, which is called induction. Newton was indebted
to it for his theorem of the binomial and the principle of
universal gravity. – LAPLACE v
4.1  Introduction
One key basis for mathematical thinking is deductive rea-
soning. An informal, and example of deductive reasoning,
borrowed from the study of logic, is an argument expressed
in three statements:
(a) Socrates is a man.
(b) All men are mortal, therefore,
(c) Socrates is mortal.
If statements (a) and (b) are true, then the truth of (c) is
established. To make this simple mathematical example,
we could write:
(i) Eight is divisible by two.
(ii) Any number divisible by two is an even number,
therefore,
(iii) Eight is an even number.
Thus, deduction in a nutshell is given a statement to be proven, often called a
conjecture or a theorem in mathematics, valid deductive steps are derived and a
proof may or may not be established, i.e., deduction is the application of a general
case to a particular case.
In contrast to deduction, inductive reasoning depends on working with each case,
and developing a conjecture by observing incidences till we have observed each and
every case. It is frequently used in mathematics and is a key aspect of scientific
reasoning, where collecting and analysing data is the norm. Thus, in simple language,
we can say the word induction means the generalisation from particular cases or facts.
Chapter
4
PRINCIPLE OF
MATHEMATICAL INDUCTION
G . Peano
(1858-1932)
2022-23
PRINCIPLE OF MATHEMATICAL INDUCTION       87
In algebra or in other discipline of mathematics, there are certain results or state-
ments that are formulated in terms of n, where n is a positive integer. To prove such
statements the well-suited principle that is used–based on the specific technique, is
known as the principle of mathematical induction.
4.2  Motivation
In mathematics, we use a form of complete induction called mathematical induction.
To understand the basic principles of mathematical induction, suppose a set of thin
rectangular tiles are placed as shown in Fig 4.1.
Fig 4.1
When the first tile is pushed in the indicated direction, all the tiles will fall. To be
absolutely sure that all the tiles will fall, it is sufficient to know that
(a) The first tile falls, and
(b) In the event that any tile falls its successor necessarily falls.
This is the underlying principle of mathematical induction.
We know, the set of natural numbers N is a special ordered subset of the real
numbers. In fact, N is the smallest subset of R with the following property:
A set S is said to be an inductive set if 1? S and  x + 1 ? S whenever x ? S. Since
N is the smallest subset of R which is an inductive set, it follows that any subset of R
that is an inductive set must contain N.
Illustration
Suppose we wish to find the formula for the sum of positive integers 1, 2, 3,...,n, that is,
a formula which will give the value of 1 + 2 + 3 when n = 3, the value 1 + 2 + 3 + 4,
when n = 4 and so on and suppose that in some manner we are led to believe that the
formula 1 + 2 + 3+...+ n = 
( 1)
2
n n +
 is the correct one.
How can this formula actually be proved? We can, of course, verify the statement
for as many positive integral values of n as we like, but this process will not prove the
formula for all values of n. What is needed is some kind of chain reaction which will
2022-23
88       MATHEMATICS
have the effect that once the formula is proved for a particular positive integer the
formula will automatically follow for the next positive integer and the next  indefinitely.
Such a reaction may be considered as produced by the method of mathematical induction.
4.3  The Principle of Mathematical Induction
Suppose there is a given statement P(n)  involving the natural number n such that
(i) The statement is true for n = 1, i.e., P(1) is true, and
(ii) If the statement is true for n = k (where k is some positive integer), then
the statement is also true for n = k + 1, i.e., truth of P(k) implies the
truth of P (k + 1).
Then, P(n) is true for all natural numbers n.
Property (i) is simply  a statement of fact. There may be situations when a
statement is true for all n = 4. In this case, step 1 will start from n = 4 and we shall
verify the result for n = 4, i.e., P(4).
Property (ii) is a conditional property. It does not assert that the given statement
is true for n = k, but only that if it is true for n = k, then it is also true for n = k +1. So,
to prove that the  property holds , only prove that conditional proposition:
If the statement is true for n = k, then it is also true for n = k + 1.
This is sometimes referred to as the inductive step. The assumption that the given
statement is true for n = k in this inductive step is called the inductive hypothesis.
For example, frequently in mathematics, a formula will be discovered that appears
to fit a pattern like
1 = 1
2   
=1
4 = 2
2
 = 1 + 3
9 = 3
2
 = 1 + 3 + 5
16 = 4
2
 = 1 + 3 + 5 + 7, etc.
It is worth to be noted that the sum of the first two odd natural numbers is the
square of second natural number, sum of the first three odd natural numbers is the
square of third natural number and so on.Thus, from this pattern it appears that
1 + 3 + 5 + 7 + ... + (2n – 1) = n
2 
, i.e,
the sum of the first n odd natural numbers is the square of n.
Let us write
P(n): 1 + 3 + 5 + 7 + ... + (2n – 1) = n
2
.
We wish to prove that P(n) is true for all n.
The first step in a proof that uses mathematical induction is to prove that
P (1) is true. This step is called the basic step. Obviously
1 = 1
2
, i.e., P(1) is true.
The next step is called the inductive step. Here, we suppose that P (k) is true for some
2022-23
PRINCIPLE OF MATHEMATICAL INDUCTION       89
positive integer k and we need to prove that P (k + 1) is true. Since P (k) is true, we
have
1 + 3 + 5 + 7 + ... + (2k – 1) = k
2
... (1)
Consider
1 + 3 + 5 + 7 + ... + (2k – 1) + {2(k +1) – 1} ... (2)
= k
2 
+ (2k + 1) = (k + 1)
2
 [Using (1)]
Therefore, P (k + 1) is true and the inductive proof is now completed.
Hence P(n) is true for all natural numbers n.
Example 1 For all n = 1, prove that
1
2 
+ 2
2 
+ 3
2 
+ 4
2 
+…+ n
2     
= 
( 1)(2 1)
6
n n n + +
.
Solution Let the given statement be P(n), i.e.,
P(n) :  1
2 
+ 2
2 
+ 3
2 
+ 4
2 
+…+ n
2     
= 
( 1)(2 1)
6
n n n + +
For n = 1, P(1): 1 = 
1(1 1)(2 1 1)
6
+ × +
=  
1 2 3
1
6
× ×
=
 which is true.
Assume that P(k) is true for some positive integer k, i.e.,
1
2 
+ 2
2 
+ 3
2 
+ 4
2 
+…+ k
2     
= 
( 1)(2 1)
6
k k k + +
... (1)
We shall now prove that P(k + 1) is also true. Now, we have
(1
2
  +2
2
  +3
2
  +4
2
  +…+k
2
  ) + (k + 1)
 2
=
2
( 1)(2 1)
( 1)
6
k k k
k
+ +
+ + [Using (1)]
=
2
( 1)(2 1) 6( 1)
6
k k k k + + + +
=
2
( 1)(2 7 6)
6
k k k + + +
=
( 1)( 1 1){2( 1) 1 }
6
k k k + + + + +
Thus P(k + 1) is true, whenever P (k) is true.
Hence, from the principle of mathematical induction, the statement P(n) is true
for all natural numbers n.
2022-23
Page 5


vAnalysis and natural philosophy owe their most important discoveries to
this fruitful means, which is called induction. Newton was indebted
to it for his theorem of the binomial and the principle of
universal gravity. – LAPLACE v
4.1  Introduction
One key basis for mathematical thinking is deductive rea-
soning. An informal, and example of deductive reasoning,
borrowed from the study of logic, is an argument expressed
in three statements:
(a) Socrates is a man.
(b) All men are mortal, therefore,
(c) Socrates is mortal.
If statements (a) and (b) are true, then the truth of (c) is
established. To make this simple mathematical example,
we could write:
(i) Eight is divisible by two.
(ii) Any number divisible by two is an even number,
therefore,
(iii) Eight is an even number.
Thus, deduction in a nutshell is given a statement to be proven, often called a
conjecture or a theorem in mathematics, valid deductive steps are derived and a
proof may or may not be established, i.e., deduction is the application of a general
case to a particular case.
In contrast to deduction, inductive reasoning depends on working with each case,
and developing a conjecture by observing incidences till we have observed each and
every case. It is frequently used in mathematics and is a key aspect of scientific
reasoning, where collecting and analysing data is the norm. Thus, in simple language,
we can say the word induction means the generalisation from particular cases or facts.
Chapter
4
PRINCIPLE OF
MATHEMATICAL INDUCTION
G . Peano
(1858-1932)
2022-23
PRINCIPLE OF MATHEMATICAL INDUCTION       87
In algebra or in other discipline of mathematics, there are certain results or state-
ments that are formulated in terms of n, where n is a positive integer. To prove such
statements the well-suited principle that is used–based on the specific technique, is
known as the principle of mathematical induction.
4.2  Motivation
In mathematics, we use a form of complete induction called mathematical induction.
To understand the basic principles of mathematical induction, suppose a set of thin
rectangular tiles are placed as shown in Fig 4.1.
Fig 4.1
When the first tile is pushed in the indicated direction, all the tiles will fall. To be
absolutely sure that all the tiles will fall, it is sufficient to know that
(a) The first tile falls, and
(b) In the event that any tile falls its successor necessarily falls.
This is the underlying principle of mathematical induction.
We know, the set of natural numbers N is a special ordered subset of the real
numbers. In fact, N is the smallest subset of R with the following property:
A set S is said to be an inductive set if 1? S and  x + 1 ? S whenever x ? S. Since
N is the smallest subset of R which is an inductive set, it follows that any subset of R
that is an inductive set must contain N.
Illustration
Suppose we wish to find the formula for the sum of positive integers 1, 2, 3,...,n, that is,
a formula which will give the value of 1 + 2 + 3 when n = 3, the value 1 + 2 + 3 + 4,
when n = 4 and so on and suppose that in some manner we are led to believe that the
formula 1 + 2 + 3+...+ n = 
( 1)
2
n n +
 is the correct one.
How can this formula actually be proved? We can, of course, verify the statement
for as many positive integral values of n as we like, but this process will not prove the
formula for all values of n. What is needed is some kind of chain reaction which will
2022-23
88       MATHEMATICS
have the effect that once the formula is proved for a particular positive integer the
formula will automatically follow for the next positive integer and the next  indefinitely.
Such a reaction may be considered as produced by the method of mathematical induction.
4.3  The Principle of Mathematical Induction
Suppose there is a given statement P(n)  involving the natural number n such that
(i) The statement is true for n = 1, i.e., P(1) is true, and
(ii) If the statement is true for n = k (where k is some positive integer), then
the statement is also true for n = k + 1, i.e., truth of P(k) implies the
truth of P (k + 1).
Then, P(n) is true for all natural numbers n.
Property (i) is simply  a statement of fact. There may be situations when a
statement is true for all n = 4. In this case, step 1 will start from n = 4 and we shall
verify the result for n = 4, i.e., P(4).
Property (ii) is a conditional property. It does not assert that the given statement
is true for n = k, but only that if it is true for n = k, then it is also true for n = k +1. So,
to prove that the  property holds , only prove that conditional proposition:
If the statement is true for n = k, then it is also true for n = k + 1.
This is sometimes referred to as the inductive step. The assumption that the given
statement is true for n = k in this inductive step is called the inductive hypothesis.
For example, frequently in mathematics, a formula will be discovered that appears
to fit a pattern like
1 = 1
2   
=1
4 = 2
2
 = 1 + 3
9 = 3
2
 = 1 + 3 + 5
16 = 4
2
 = 1 + 3 + 5 + 7, etc.
It is worth to be noted that the sum of the first two odd natural numbers is the
square of second natural number, sum of the first three odd natural numbers is the
square of third natural number and so on.Thus, from this pattern it appears that
1 + 3 + 5 + 7 + ... + (2n – 1) = n
2 
, i.e,
the sum of the first n odd natural numbers is the square of n.
Let us write
P(n): 1 + 3 + 5 + 7 + ... + (2n – 1) = n
2
.
We wish to prove that P(n) is true for all n.
The first step in a proof that uses mathematical induction is to prove that
P (1) is true. This step is called the basic step. Obviously
1 = 1
2
, i.e., P(1) is true.
The next step is called the inductive step. Here, we suppose that P (k) is true for some
2022-23
PRINCIPLE OF MATHEMATICAL INDUCTION       89
positive integer k and we need to prove that P (k + 1) is true. Since P (k) is true, we
have
1 + 3 + 5 + 7 + ... + (2k – 1) = k
2
... (1)
Consider
1 + 3 + 5 + 7 + ... + (2k – 1) + {2(k +1) – 1} ... (2)
= k
2 
+ (2k + 1) = (k + 1)
2
 [Using (1)]
Therefore, P (k + 1) is true and the inductive proof is now completed.
Hence P(n) is true for all natural numbers n.
Example 1 For all n = 1, prove that
1
2 
+ 2
2 
+ 3
2 
+ 4
2 
+…+ n
2     
= 
( 1)(2 1)
6
n n n + +
.
Solution Let the given statement be P(n), i.e.,
P(n) :  1
2 
+ 2
2 
+ 3
2 
+ 4
2 
+…+ n
2     
= 
( 1)(2 1)
6
n n n + +
For n = 1, P(1): 1 = 
1(1 1)(2 1 1)
6
+ × +
=  
1 2 3
1
6
× ×
=
 which is true.
Assume that P(k) is true for some positive integer k, i.e.,
1
2 
+ 2
2 
+ 3
2 
+ 4
2 
+…+ k
2     
= 
( 1)(2 1)
6
k k k + +
... (1)
We shall now prove that P(k + 1) is also true. Now, we have
(1
2
  +2
2
  +3
2
  +4
2
  +…+k
2
  ) + (k + 1)
 2
=
2
( 1)(2 1)
( 1)
6
k k k
k
+ +
+ + [Using (1)]
=
2
( 1)(2 1) 6( 1)
6
k k k k + + + +
=
2
( 1)(2 7 6)
6
k k k + + +
=
( 1)( 1 1){2( 1) 1 }
6
k k k + + + + +
Thus P(k + 1) is true, whenever P (k) is true.
Hence, from the principle of mathematical induction, the statement P(n) is true
for all natural numbers n.
2022-23
90       MATHEMATICS
Example 2 Prove that 2
n 
> n for all positive integers n.
Solution Let P(n):  2
n
 > n
When n =1, 2
1 
>1. Hence P(1) is true.
Assume that P(k) is true for any positive integer k, i.e.,
2
k 
 > k ... (1)
We shall now prove that P(k +1) is true whenever P(k) is true.
Multiplying both sides of (1) by 2, we get
2. 2
k
 
 > 2k
i.e., 2 
k + 1
 > 2k = k + k > k + 1
Therefore, P(k + 1) is true when P(k) is true. Hence, by principle of mathematical
induction, P(n) is true for every positive integer n.
Example 3 For all n = 1, prove that
1 1 1 1
...
1.2 2.3 3.4 ( 1) 1
n
n n n
+ + + + =
+ +
.
Solution We can write
P(n):  
1 1 1 1
...
1.2 2.3 3.4 ( 1) 1
n
n n n
+ + + + =
+ +
We note that P(1):
1 1 1
1.2 2 1 1
= =
+
, which is true. Thus, P(n) is true for n = 1.
Assume that P(k) is true for some natural number k,
i.e.,
1 1 1 1
...
1.2 2.3 3.4 ( 1) 1
k
k k k
+ + + + =
+ +
... (1)
We need to prove that P(k + 1) is true whenever P(k) is true. We have
1 1 1 1 1
...
1.2 2.3 3.4 ( 1) ( 1) ( 2) k k k k
+ + + + +
+ + +
= 
1 1 1 1 1
...
1.2 2.3 3.4 ( 1) ( 1)( 2) k k k k
? ?
+ + + + +
? ?
+ + +
? ?
=
1
1 ( 1)( 2)
k
k k k
+
+ + +
[Using (1)]
2022-23