IST 230- Lesson 6 Quiz Sequences & Induction - Write the first four terms of the sequence defined by the formula:

Question # 00080415 Posted By: jia_andy Updated on: 07/07/2015 11:04 AM Due on: 11/25/2015
Subject Mathematics Topic General Mathematics Tutorials:
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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

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)



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