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.

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

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

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

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.

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.