Question

Please help me in finding RHS by using Principle of Mathematical Induction...​

Answer 1

Answer:

What is mathematical Induction???

Please tell

Answer 2

Answer:

Mathematical induction is a mathematical proof technique used to prove a given statement about any well-ordered set. Most commonly, it is used to establish statements for the set of all natural numbers.

Mathematical induction is a form of direct proof, usually done in two steps. When trying to prove a given statement for a set of natural numbers, the first step, known as the base case, is to prove the given statement for the first natural number.

i.e Prove that the statement P(n) is true for n=1.

The second step, known as the inductive step, is to prove that, if the statement is assumed to be true for any one natural number, then it must be true for the next natural number as well.

i.e Assume that the statement P(n) is true for n=k and then prove that it is equally true for n=k+1.

Having proved these two steps, the rule of inference establishes the statement to be true for all natural numbers. In common terminology, using the stated approach is referred to as using the Principle of mathematical induction.

Step-by-step explanation:

Mathematical induction can be used to prove that the following statement, P(n), holds for all natural numbers n.

P(n) gives a formula for the sum of the natural numbers less than or equal to number n. The proof that P(n) is true for each natural number n proceeds as follows.

Basis: Show that the statement holds for n = 0.

P(0) amounts to the statement:

In the left-hand side of the equation, the only term is 0, and so the left-hand side is simply equal to 0.

In the right-hand side of the equation, 0·(0 + 1)/2 = 0.

The two sides are equal, so the statement is true for n = 0. Thus it has been shown that P(0) holds.

Inductive step: Show that if P(k) holds, then also P(k + 1) holds. This can be done as follows.

Assume P(k) holds (for some unspecified value of k). It must then be shown that P(k + 1) holds, that is:

Using the induction hypothesis that P(k) holds, the left-hand side can be rewritten to:

thereby showing that indeed P(k + 1) holds.

Since both the basis and the inductive step have been performed, by mathematical induction, the statement P(n) holds for all natural numbers n. Q.E.D.