Question

DISCRETE MATHEMATICS PROOF PROBLEMS

1. Use a proof by induction to show that, −(16 − 11?) is a positive number that is divisible by 5 when ? ≥ 2.

2.Prove (using a formal proof technique) that any sequence that begins with the first four integers 12, 6, 4, 3 is neither arithmetic, nor geometric.

Answer 1

1.

2. In an arithmetic sequence the difference between one term and the next is a constant. The terms of the sequence is of the form {a, a+d,a+2d,....}.

Here, the sequence begins with 12,6,4,3.

6 - 12 = -6

4 - 6 = -2

3 - 4 = -1

-6 ≠ -2 ≠ -1

Hence there is no common difference and therefore the sequence is not arithmetic.

A geometric sequence has a common ratio.

Here, 6/12 = 1/2

4/6. = 2/3 ≠ 1/2

Hence there is no common ratio and hence is not geometric series.