IST 230 Lesson 6 Quiz Sequences & Induction  Write the first four terms of the sequence defined by the formula:
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) = 2n^{2}
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 n^{2} + 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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Solution: IST 230 Lesson 6 Quiz Sequences & Induction  Write the first four terms of the sequence defined by the formula: