Proof that by mathematical induction
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Answer:
Proofs by Induction A proof by induction is just like an ordinary proof in which every step must be justified. However it employs a neat trick which allows you to prove a statement about an arbitrary number n by first proving it is true when n is 1 and then assuming it is true for n=k and showing it is true for n=k+1.
Answer:
Using the principle to proof by mathematical induction we need to follow the techniques and steps exactly as shown.
We note that a prove by mathematical induction consists of three steps.
• Step 1. (Basis) Show that P(n₀) is true.
• Step 2. (Inductive hypothesis). Write the inductive hypothesis: Let k be an integer such that k ≥ n₀ and P(k) be true.
• Step 3. (Inductive step). Show that P(k + 1) is true.
In mathematical induction we can prove an equation statement where infinite number of natural numbers exists but we don’t have to prove it for every separate numbers. We use only two steps to prove it namely base step and inductive step to prove the whole statement for all the cases. Practically it’s not possible to prove a mathematical statement or formula or equation for all the natural numbers but we can generalize the statement by proving with induction method. As if the statement is true for P (k), it will be true for P (k+1), so if it is true for P (1) then it can be proved for P (1+1) or P (2) similarly for P (3), P (4) and so on up to n natural numbers.
In Proof by mathematical induction the first principle is if the base step and inductive step are proved then P (n) is true for all natural numbers. In inductive step we need to assume P (k) is true and this assumption is called as induction hypothesis. By using this assumption we prove P (k+1) is true. While proving for the base case we can take P (0) or P (1).
Step-by-step explanation: