Exercise · ZW-DIRECT-PROOF-002 · ★☆☆ Basic
If \(n=2k+1\), what is \(n^2\) equal to?
Choose one option.
Show answer
- C. \(4k^2+4k+1\)
Explanation
Expand the square: \((2k+1)^2=4k^2+4k+1=2(2k^2+2k)+1\), which is odd.
Review the concept: Direct proof
A direct proof of an implication \(p\Rightarrow q\) starts from the hypothesis \(p\) and, using definitions and previously proved results, chains steps until it reaches the conclusion \(q\).
\[p\Rightarrow\cdots\Rightarrow q\]
More practice problems: direct proof
- ZW-DIRECT-PROOF-001 To prove directly that if n is even then n^2 is even, one starts by writing:
- ZW-DIRECT-PROOF-003 Let a=2k+1 and b=2j+1 be odd. Which expression shows that a+b is even?
- ZW-DIRECT-PROOF-004 In a direct proof of p⇒ q, what do we do?
- ZW-DIRECT-PROOF-005 Which of these starts is a direct proof of «if n is odd then n+1 is even»?
- ZW-DIRECT-PROOF-006 Let a=2x and b=2y be even. What follows about ab?
- ZW-DIRECT-PROOF-007 We know a| b and b| c, i.e. b=ak and c=bm. How do we conclude a| c?
- ZW-DIRECT-PROOF-008 Does checking that 3^2=9, 5^2=25 and 7^2=49 are odd prove that the square of every odd number is od…
- ZW-DIRECT-PROOF-009 To prove that if n is odd then 8| n^2-1, write n=2k+1. What is n^2-1 equal to?