IST 230 (Show/Explain all Work) Lesson 6 Quiz Sequences & Induction 1. Write the first four terms of the sequence defined by the formula: 2. Give an explicit formula for the sequence: 3. (This Problem Counts Double) 4. Given the following original summation: Write the original summation in expanded form: Separate off the final term from the original summation: 5. Write the following sum using summation (sigma) notation: 6. Give a simpler but equivalent expression for 7. Give the standard recursive definition of n! for n 0: 8. Transform the following expression by making the change of variable x = j + 2: 9. Use the formula for the sum of the first n integers to calculate: 5+6+7+...+15 Larry Newcomerqattachments_b23374242e9a33bfa39b576c82d3375d950036a8 (16 @ 6 Each) Page 1 of 2 The Online Course Content and textbook are the ONLY sources of help that may be used for this exam IST 230 (Show/Explain all Work) Lesson 6 Quiz Sequences & Induction 10. Use the formula for the sum of a geometric sequence to calculate: HINT: 11. (THIS PROBLEM COUNTS DOUBLE) 12. Prove the following conjecture using Mathematical Induction: For all integers n 1: 2 + 6+ 10 +...+ (4n - 2) = 2n2 13. (THIS PROBLEM COUNTS DOUBLE) 14. Prove the following conjecture using Mathematical Induction (ALTHOUGH YOU MAY USE COMMUTATIVITY, DO NOT USE ONLY ALGEBRA AND THE COMMUTATIVE LAW FOR MULTIPLICATION - YOU MUST GIVE AN INDUCTIVE PROOF TO GET ANY CREDIT): 15. (THIS PROBLEM COUNTS DOUBLE) 16. Using Mathematical Induction, prove that n2 + n is even for all integers n >= 1 (Remember, an integer x is even if there exists an integer k such that x = 2k) Larry Newcomerqattachments_b23374242e9a33bfa39b576c82d3375d950036a8 (16 @ 6 Each) Page 2 of 2 The Online Course Content and textbook are the ONLY sources of help that may be used for this exam