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
Question
- 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
Solution
The purpose of the base case in mathematical induction is c) To provide a starting point for the induction.
Mathematical induction is a method of mathematical proof typically used to establish a given statement for all natural numbers. It is a two-step process:
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Base Case: This is the starting point of the induction. It is the case where the statement is checked for the initial value (usually n=1 or n=0). The base case serves to verify that the statement holds for the initial value.
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Inductive Step: This is where the property is proven for a general case, often denoted as 'k'. Then it is shown that if the property holds for the case 'k', it also holds for the case 'k+1'. This step is crucial as it ensures that the property can be extended to all natural numbers.
So, the base case is essential as it provides the starting point from which the inductive step can proceed.
Similar Questions
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
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
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.
14) What is the primary goal of mathematical induction?a) To derive the quadratic formulab) To calculate areas under curvesc) To prove statements for all natural numbersd) To solve algebraic 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.
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