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17) Why is the inductive hypothesis important in mathematical induction?a) It calculates numerical valuesb) It sovles for unknown variablesc) It provides a basis for proving the inductive stepd) It disproves the given statement

Question

  1. Why is the inductive hypothesis important in mathematical induction?a) It calculates numerical valuesb) It sovles for unknown variablesc) It provides a basis for proving the inductive stepd) It disproves the given statement
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Solution

The inductive hypothesis is important in mathematical induction because of option c) It provides a basis for proving the inductive step.

Here's a step-by-step explanation:

  1. Mathematical induction is a method of mathematical proof typically used to establish a given statement for all natural numbers. It is a form of direct proof, and it is done in two steps: the base case and the inductive step.

  2. The base case (or basis) proves the statement for the first natural number, usually 1 or 0.

  3. The inductive step is where the inductive hypothesis comes into play. The inductive hypothesis is an assumption that the statement is true for some natural number n.

  4. Then, you prove that the statement holds for the next natural number n+1, under the assumption that it holds for n. This is done by manipulating the inductive hypothesis to show that if it's true, then the statement for n+1 must also be true.

  5. This process creates a chain reaction: if the statement is true for 1, then it must be true for 2, if it's true for 2, then it must be true for 3, and so on. This is why the inductive hypothesis is so important - it's the foundation of this chain reaction.

So, the inductive hypothesis doesn't calculate numerical values or solve for unknown variables, and it certainly doesn't disprove the given statement. Instead, it provides a basis for proving the inductive step, which is crucial for the method of mathematical induction.

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Similar Questions

15) In mathematical induction, what is the purpose of the base case?a) To provide a general formulab) To disprove the given statementc) To provide a starting point for the inductiond) To solve for unknown variables

16) What is proven in the inductive step of mathematical induction?a) The statement holds for all real numbers.b) The statement holds for an arbitrary case.c) The statement holds for the next case, given it holds for a particular case.d) The statement holds for the base case.

18) Why might a mathematician choose to use mathematical induction to prove a statement?a) To prove the statement for an infinite number of casesb) To find the roots of a polynomialc) To disprove a conjectured) To solve a system of equations

1. Let P (n) be the statement that  2 2 2 2 1 2 11 2 3 ... 6n n nn       for the positiveinteger n.a) What is the statement P (1)?b) Show that P (1) is true, completing the basis step of the proof.c) What is the inductive hypothesis?d) What do you need to prove in the inductive step?e) Complete the inductive step, identifying where you use the inductive hypothesis.

2. Let P (n) be the statement that21 1 1 11 ... 24 9 n n      , where n is an integer greaterthan 1.a) What is the statement P (2)?b) Show that P (2) is true, completing the basis step of the proof.c) What is the inductive hypothesis?d) What do you need to prove in the inductive step?e) Complete the inductive step.

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