Simple deductive proof practice questions.

In a deductive proof, which expression correctly represents a general even integer?

• 2k + 1, where k is an integer
• k + 2, where k is an integer
• k², where k is an integer
• 2k, where k is an integer

Answer: 2k, where k is an integer

Every even integer is a multiple of 2, so it is written 2k with k an integer; 2k + 1 represents an odd integer, and k + 2 has the same parity as k.

Which of the following is a valid deductive method for proving that an algebraic identity holds?

• Start from one side and transform it algebraically, step by step, into the other side
• Substitute several values of the variable and check that both sides agree
• Assume the identity is true and simplify both sides until reaching 0 = 0
• Plot both sides with technology and observe that the graphs coincide

Answer: Start from one side and transform it algebraically, step by step, into the other side

A deductive proof must follow logically from accepted rules for all values, which working from one side to the other achieves; checking examples or assuming the result does not establish general truth.

For an integer n, which pair of expressions represents two consecutive odd integers?

• 2n and 2n + 2
• 2n + 1 and 2n + 2
• 2n + 1 and 2n + 3
• n and n + 1

Answer: 2n + 1 and 2n + 3

2n + 1 is odd for any integer n, and the next odd integer is two more: 2n + 3. Adding only 1 gives an even number, and 2n, 2n + 2 are consecutive even integers.

As part of a proof, a student expands (n + 3)² − (n − 3)², where n is an integer. Which simplified result is correct?

• 6n
• 12n
• 12n + 18
• 0

Answer: 12n

(n² + 6n + 9) − (n² − 6n + 9) = 12n; the n² and constant terms cancel while the middle terms combine. Forgetting that the constant 9 cancels gives the +18 distractor.

In the identity (x + 1)² ≡ x² + 2x + 1, what does the symbol ≡ indicate?

• The two expressions are equal for every value of the variable
• The two expressions are approximately equal
• The two expressions are equal for exactly one value of the variable
• The two expressions are equal only for positive values of the variable

Answer: The two expressions are equal for every value of the variable

The identity symbol ≡ states that both sides are identically equal — equal for all values of x — unlike an equation, which may hold only for particular solutions.

To prove that the product of two even integers is divisible by 4, the integers are written as 2m and 2n. Which expression for their product completes the proof?

• 2mn
• 4(m + n)
• 2(m + n)
• 4mn

Answer: 4mn

2m × 2n = 4mn, and since mn is an integer, the product is a multiple of 4. Multiplying, not adding, the two factors is the essential step.