Question

Prove that the following statement is true: For any odd integer n, then n²/4=(n²+3)/4. Write your proof:

Answer 1

To prove the statement n²/4 = (n²+3)/4 for any odd integer n, we can simplify both sides of the equation and show that they are equal.

Step-by-step explanation:

To prove the statement, we need to show that for any odd integer n, the equation n²/4 = (n²+3)/4 holds true.

Starting with the left side of the equation, n²/4, we can simplify it by dividing n² by 4, which gives us n²/4.

Now, let's simplify the right side of the equation, (n²+3)/4. We can expand the numerator by adding 3 to n², which results in (n²+3). Then, we divide this sum by 4, which gives us (n²+3)/4.

Since the left and right sides of the equation simplify to the same expression, n²/4 = (n²+3)/4, we have proved that the statement is true for any odd integer n.

Answer 2

The proof shows that for any odd integer n, n²/4 is equal to (n²+3)/4 by expressing n as 2k+1, squaring it, and comparing both expressions after dividing by 4.

Step-by-step explanation:

The assertion to prove is: for any odd integer n, n²/4 is equal to (n²+3)/4. Let's start by analyzing the expression on the right side of the statement.

• Consider an odd integer n; by definition, it can be expressed as n = 2k+1, where k is an integer.
• Now, we square n to obtain n² = (2k+1)² = 4k² + 4k + 1.
• To validate the initial statement, we divide n² by 4, which yields (4k² + 4k + 1)/4 = k² + k + 1/4.
• Add 3 to n² and then divide by 4, we get (n² + 3)/4 = (4k² + 4k + 4)/4 = k² + k + 1.

Upon comparing the two results, we observe that they are identical. This demonstrates that for any odd integer n, n²/4 indeed equals (n²+3)/4.