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Which of the following is a valid proof for the statement: If n is an even integ
Which of the following is a valid proof for the statement: If n is an even integer, then n^2 is even?
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Practice Questions
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Q1
Which of the following is a valid proof for the statement: If n is an even integer, then n^2 is even?
n = 2k for some integer k
n = 2k + 1
n = k^2
n = 2k - 1
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If n = 2k, then n^2 = (2k)^2 = 4k^2, which is even.
Questions & Step-by-step Solutions
1 item
Q
Q: Which of the following is a valid proof for the statement: If n is an even integer, then n^2 is even?
Solution:
If n = 2k, then n^2 = (2k)^2 = 4k^2, which is even.
Steps: 5
Show Steps
Step 1: Understand what an even integer is. An even integer can be written as n = 2k, where k is any integer.
Step 2: Substitute n in the expression n^2. If n = 2k, then n^2 = (2k)^2.
Step 3: Calculate (2k)^2. This equals 4k^2.
Step 4: Recognize that 4k^2 is even. Since 4 is an even number and multiplying it by any integer (k^2) will also result in an even number.
Step 5: Conclude that if n is an even integer, then n^2 is also even.
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