# Practice for First Second Exam

### Directions:

• Answer the problems on the exam pages.
• There are four problems for 100 total points. Scale will be determined after the exam but a good guess is A = 90, C = 60.
• If you need extra space use the back of a page.
• No books, notes, calculators, or collaboration.

```  Q1: 15 points
Q2: 15 points
Q3: 20 points
Q4: 20 points
Q5: 30 points
Total: 100 points
```

Corrections in green made on 20 October 2015.

• Question 1 (15): Briefly identify the following terms or concepts (3 points each):

• (a) for x to be a multiple of y

• (b) a proof by contradiction

• (c) a rational number

• (d) the base case of a proof by induction

• (e) a predicate on the positive integers

• Question 2 (15): Determine whether each of these five statements is true or false. No explanation is needed or wanted, and there is no penalty for guessing.

• (a) 91 is not divisible by 7.

• (b) (31 mod 6) = (25 mod 4) (Remember that E&C's "mod" is the same as Java's "%" on non-negative integers.)

• (c) The binary representation of the decimal number "19" is "10101".

• (d) Every perfect square can be written as either "4k" or "4k+1" for some integer k.

• (e) The square of every irrational number is irrational.

• Question 3 (20): Prove that if k is any integer, the number 7k + 5 is not a perfect square. (Hint: Argue the contrapositive, and apply the Division Theorem with m = 7 to the number you are squaring.)

• Question 4 (20): Define a number sequence so that a1 = 3 and for all k with k > 1, ak = ak-1 + 4k - 7. Prove by induction that for any positive integer n, an = 2n2 - 5n + 6.

• Question 5 (30): Define a number sequence so that b1 = 2, b2 = 3, and for all k with k > 2, bk = bk-1 - bk-2 + 1.

• (a, 5) Calculate bi for all integers i from i through 10.

• (b, 5) Determine the value of b100. (Hint: I don't expect you to determine all the intermediate values -- find a pattern in part (a) or use the result of part (c).)

• (c, 20) Prove by induction that for any n with n > 6, bn = bn-6. (Hint: Prove the base cases of n = 7 and n = 8, then use the sequence definition in your inductive case.)