pls explain the principle of mathematical.
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The Principle of Mathematical Induction :-
As a mathematical technique of proving things, mathematical induction is essentially used to prove the property of natural numbers (n). According to the principle of mathematical induction, a property X(n) holds to be same for all natural numbers, 0,1,2,3…… n. Let’s consider a given statement X(n) involving natural number n, so that
The statement is true for n = 1 i.e., X(1) is true, andIf the statement is true for n=k, where k is a positive integer, then the statement is also true for all cases of n= k+1 i.e., X(k) leads to the truth of X(k+1).
This means that X(n) is true for all the natural numbers, n.
Property a) mentioned above is simply a statement of a fact. In situations, where the statement is also true for all the cases of n≥5, then the step a) shall start from n=5 and we shall hence verify the result for n=5, i.e., X(5).Property b) is a conditional property as it does not confirm that the given statement is true for n=k, but if it is true for n=k then it shall also be true for n=k+1.
Therefore to prove a property we need to first prove the conditional proposition. Proving a conditional proposition is referred to as inductive step of a theorem. The presumption that the given statement, is true, in this case, n=k is termed as the inductive hypothesis.
Example
Mathematics follows a pattern, one of the many patterns is given below
1 = 12 = 1
4 = 22= 1 + 3
9 = 32 = 1 + 3 + 5
16 = 42 = 1 + 3 + 5 + 7
25 = 52 = 1 + 3 + 5 + 7 + 9
From the above pattern example, we come to a conclusion that the square of the second natural number is equal to the sum of first two odd natural numbers. Likewise, the sum of first three odd natural numbers is equal to the square of the third natural number. From the above pattern we conclude that,
1 + 3 + 5 + 7 + … + (2n – 1) = n2 , or the square of n is equal to the sum of first n odd natural numbers. Now let us write this as,
X(n) = 1 + 3 + 5 + 7 + 9 +…….+ (2n-1) = n2
With the help of the principle of mathematical induction, we need to prove that X(n) is true for all the values of n. The first step in this process is to prove the value X(1) is true. This first step is called the base step or basic step, as it forms the basis of mathematical induction. 1 = 12 , X(1) therefore is true.
The next step following the base is called the inductive step. In this step we suppose that X(m) is true for any positive integer m, we also need to prove that X(m+1) is true. Since X(m) is true we have,
1 + 3 + 5 + 7…. + (2m-1) = m2 (1)
Now consider, 1 + 3 + 5 + 7 +….+ (2m-1) + {2(m+1)-1} (2)
Using (1), we get
= m2 + (2m + 1) = (m + 1)2
Therefore, X(m + 1) is also true thus completing our inductive proof step. This gives us that X(n) is true for all the natural numbers. Through the above example, we have proved that X(n) is true for all natural numbers n.
Hope it's help you!
The Principle of Mathematical Induction :-
As a mathematical technique of proving things, mathematical induction is essentially used to prove the property of natural numbers (n). According to the principle of mathematical induction, a property X(n) holds to be same for all natural numbers, 0,1,2,3…… n. Let’s consider a given statement X(n) involving natural number n, so that
The statement is true for n = 1 i.e., X(1) is true, andIf the statement is true for n=k, where k is a positive integer, then the statement is also true for all cases of n= k+1 i.e., X(k) leads to the truth of X(k+1).
This means that X(n) is true for all the natural numbers, n.
Property a) mentioned above is simply a statement of a fact. In situations, where the statement is also true for all the cases of n≥5, then the step a) shall start from n=5 and we shall hence verify the result for n=5, i.e., X(5).Property b) is a conditional property as it does not confirm that the given statement is true for n=k, but if it is true for n=k then it shall also be true for n=k+1.
Therefore to prove a property we need to first prove the conditional proposition. Proving a conditional proposition is referred to as inductive step of a theorem. The presumption that the given statement, is true, in this case, n=k is termed as the inductive hypothesis.
Example
Mathematics follows a pattern, one of the many patterns is given below
1 = 12 = 1
4 = 22= 1 + 3
9 = 32 = 1 + 3 + 5
16 = 42 = 1 + 3 + 5 + 7
25 = 52 = 1 + 3 + 5 + 7 + 9
From the above pattern example, we come to a conclusion that the square of the second natural number is equal to the sum of first two odd natural numbers. Likewise, the sum of first three odd natural numbers is equal to the square of the third natural number. From the above pattern we conclude that,
1 + 3 + 5 + 7 + … + (2n – 1) = n2 , or the square of n is equal to the sum of first n odd natural numbers. Now let us write this as,
X(n) = 1 + 3 + 5 + 7 + 9 +…….+ (2n-1) = n2
With the help of the principle of mathematical induction, we need to prove that X(n) is true for all the values of n. The first step in this process is to prove the value X(1) is true. This first step is called the base step or basic step, as it forms the basis of mathematical induction. 1 = 12 , X(1) therefore is true.
The next step following the base is called the inductive step. In this step we suppose that X(m) is true for any positive integer m, we also need to prove that X(m+1) is true. Since X(m) is true we have,
1 + 3 + 5 + 7…. + (2m-1) = m2 (1)
Now consider, 1 + 3 + 5 + 7 +….+ (2m-1) + {2(m+1)-1} (2)
Using (1), we get
= m2 + (2m + 1) = (m + 1)2
Therefore, X(m + 1) is also true thus completing our inductive proof step. This gives us that X(n) is true for all the natural numbers. Through the above example, we have proved that X(n) is true for all natural numbers n.
Hope it's help you!
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