NCERT Class 11-Math's: Chapter –4 Principle of Mathematical Induction Part 1
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4.1 Overview
Mathematical induction is one of the techniques which can be used to prove variety of mathematical statements which are formulated in terms of , where is a positive integer.
4.1.1 The principle of mathematical induction
Let be a given statement involving the natural number n such that
(i) The statement is true for is true (or true for any fixed natural number) and
(ii) If the statement is true for (where is a particular but arbitrary natural number), then the statement is also true for , i.e, truth of P(k) implies the truth of . Then is true for all natural numbers .
4.2 Solved Examples
Short Answer Type
Prove statements in Examples 1 to 5, by using the Principle of Mathematical Induction for all , that :
Question 1:
Answer:
Let the given statement be defined as , for . Note that is true, since
Assume that is true for some
Now, to prove that is true, we have
Thus, is true, whenever is true.
Hence, by the Principle of Mathematical Induction, is true for all .
Question 2:
Answer:
We observe that
Thus, in true for .
Assume that is true for .
To prove that is true, we have
Thus, is true, whenever is true.
Hence, by the Principle of Mathematical Induction, is true for all natural numbers .
Question 3:
for all natural numbers, .
Answer:
Let the given statement be , i.e.,
for all natural numbers, We, observe that is true, since
Assume that is true for some , i.e.,
Now, to prove that is true, we have
Thus, is true, whenever is true.
Hence, by the Principle of Mathematical Induction, is true for all natural numbers, .