The technique that is used for proving the results or for natural numbers, the statements are established is known as mathematical induction.
A mathematical technique used for proving a statement, formula or a theorem is true for every natural number is known as Mathematical Induction. A statement can be proved in two steps:
Step 1(Base step) – The statement is proved to be true for the initial value.
Step 2(Inductive step) – The statement is proved to be true for the nth iteration.
Step 1 – An initial value for which the statement is true is considered. It is represented as n = initial value.
Step 2 – It is assumed that the statement is true for any value of n = k. Then the statement is proved true for n = k+1. Here n = k+1 is broken down into two parts, one part is n = k , which is the proved one and the second part is proved.
Step 1 − For which is a multiple of 2
Step 2 – It is assumed that3n−1 is true for n=k and hence, 3k−1 is true.
It is to be proved that
The first part (2×3k) is a multiple of 2 and the second part (3k−1) is also true as our previous assumption.
Hence it is proved that3n–1 is a multiple of 2.
Step 1 − For , Hence, step 1 is satisfied.
Step 2 – It is assumed that the statement is true for n=k.
For every natural number n, prove thatis true.
Step 1 − ForHence, step 1 is satisfied.
One more form of mathematical induction is Strong Induction. This induction proves that a propositional function P(n) is true for all the positive integers , by using the following steps:
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