Question

1. Let n be an integer. Prove that if n is even, then n^2 + 2n + 3 is odd. 2. Prove that if 2n^2 + 7n is even and n is an integer, then n is even. 3. Let n be an integer. Prove that n is odd if and only if 3n^2 - 7 is even.

          1. Let n be an integer. Prove that if n is even,
then n^2 + 2n + 3 is odd.

2. Prove that if 2n^2 + 7n is even and n is an
integer, then n is even.

3. Let n be an integer. Prove that n is odd if
and only if 3n^2 - 7 is even.

Added by Debra R.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

09/03/2023

Step 1

If $n$ is even, then $n = 2k$ for some integer $k$. Then, $n^2 + 2n + 3 = (2k)^2 + 2(2k) + 3 = 4k^2 + 4k + 3 = 2(2k^2 + 2k) + 3$. Since $2k^2 + 2k$ is an integer, let's say $2k^2 + 2k = m$. Then, $n^2 + 2n + 3 = 2m + 3$, which is odd. Therefore, if $n$ is even, Show more…

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1. Let n be an integer. Prove that if n is even, then n^2 + 2n + 3 is odd. 2. Prove that if 2n^2 + 7n is even and n is an integer, then n is even. 3. Let n be an integer. Prove that n is odd if and only if 3n^2 - 7 is even.