Process of reasoning is classified into
(i) Inductive reasoning: the conclusion is made on the basis of a particular observation or experience.
(ii) Deductive reasoning: the conclusion is made from assumptions, rather than experience. These assumptions are proved mathematically.
Induction begins by observations, and from observations we arrive at some tentative conclusions called conjectures. A conjecture may be true or false.
The principle of mathematical induction helps us in proving conjectures.
Motivation
Suppose a set of thin rectangular tiles are placed as shown in the figure,
When the first tile is pushed, all the tiles will fall. To be absolutely sure that all the tiles will fall, it is sufficient to know that
(i) the first tile falls
(ii) in the event that any tile falls its successor necessarily falls.
This is the underlying principle of mathematical induction.
A set S is said to be an inductive set if 1∈S and x+1∈S whenever x∈S.
The Principle of Mathematical Induction
Let P(n) be a statement involving natural number n, then P(n) is true for all natural numbers n if,
(i) P(1) is true
(ii) P(k+1) is true whenever P(k) is true.
i.e. to prove P(n) is true for all natural numbers n, follow two steps,
(i) verify the result for n=1.
(ii) assume the result to be true for n=k and prove the result for n=k+1.
The assumption that the given statement is true for n=k is called the inductive hypothesis.
Example: Prove by induction that, 1+5+9+...+(4n-3)=n(2n-1), for all n∈N
Let P(n) be the statement 1+5+9+...+(4n-3)=n(2n-1)
P(1) ⇒ 1=1(2*1-1)
⇒ 1=1
P(1) is true
Let P(k) be true
1+5+9+...+(4k-3)=k(2k-1)
For P(k+1) ⇒ 1+5+9+...+(4k-3)+(4(k+1)-3)=(k+1)(2(k+1)-1)
L.H.S.,
= k(2k-1)+(4k+4-3) [∵1+5+9+...+(4k-3)=k(2k-1)]
= 2k$\displaystyle \small ^{2}$ -k+4k+1
= 2k$\displaystyle \small ^{2}$ +3k+1
= 2k$\displaystyle \small ^{2}$ +2k+k+1
= 2k(k+1)+(k+1)
= (k+1)(2k+1)
= (k+1)(2(k+1)-1)
L.H.S.=R.H.S.
⇒ P(k+1) is true
Hence, by induction, P(n) is true for all n∈N
(i) Inductive reasoning: the conclusion is made on the basis of a particular observation or experience.
(ii) Deductive reasoning: the conclusion is made from assumptions, rather than experience. These assumptions are proved mathematically.
Induction begins by observations, and from observations we arrive at some tentative conclusions called conjectures. A conjecture may be true or false.
The principle of mathematical induction helps us in proving conjectures.
Motivation
Suppose a set of thin rectangular tiles are placed as shown in the figure,
(i) the first tile falls
(ii) in the event that any tile falls its successor necessarily falls.
This is the underlying principle of mathematical induction.
A set S is said to be an inductive set if 1∈S and x+1∈S whenever x∈S.
The Principle of Mathematical Induction
Let P(n) be a statement involving natural number n, then P(n) is true for all natural numbers n if,
(i) P(1) is true
(ii) P(k+1) is true whenever P(k) is true.
i.e. to prove P(n) is true for all natural numbers n, follow two steps,
(i) verify the result for n=1.
(ii) assume the result to be true for n=k and prove the result for n=k+1.
The assumption that the given statement is true for n=k is called the inductive hypothesis.
Example: Prove by induction that, 1+5+9+...+(4n-3)=n(2n-1), for all n∈N
Let P(n) be the statement 1+5+9+...+(4n-3)=n(2n-1)
P(1) ⇒ 1=1(2*1-1)
⇒ 1=1
P(1) is true
Let P(k) be true
1+5+9+...+(4k-3)=k(2k-1)
For P(k+1) ⇒ 1+5+9+...+(4k-3)+(4(k+1)-3)=(k+1)(2(k+1)-1)
L.H.S.,
= k(2k-1)+(4k+4-3) [∵1+5+9+...+(4k-3)=k(2k-1)]
= 2k$\displaystyle \small ^{2}$ -k+4k+1
= 2k$\displaystyle \small ^{2}$ +3k+1
= 2k$\displaystyle \small ^{2}$ +2k+k+1
= 2k(k+1)+(k+1)
= (k+1)(2k+1)
= (k+1)(2(k+1)-1)
L.H.S.=R.H.S.
⇒ P(k+1) is true
Hence, by induction, P(n) is true for all n∈N
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