The given expression is:

$\left\lfloor \frac{n(n+1)}{2} \right\rfloor + 1$

We need to determine the conditions on $n$ under which the expression is even or odd.

Step 1: Analyze the expression $\frac{n(n+1)}{2}$

This is the sum of the first $n$ integers. To explore the parity of the given expression, we first need to examine the value of $\frac{n(n+1)}{2}$.

• If $n(n+1)$ is divisible by 2 (which it always is, because either $n$ or $n+1$ is always even), the fraction $\frac{n(n+1)}{2}$ is an integer.

Even or Odd Behavior:

• If $n$ is even, then $n+1$ is odd, and $n(n+1)$ is divisible by 2.
• If $n$ is odd, then $n+1$ is even, and again $n(n+1)$ is divisible by 2.

Thus, $\frac{n(n+1)}{2}$ is always an integer for any $n$.

Step 2: Behavior of $\left\lfloor \frac{n(n+1)}{2} \right\rfloor$

The floor function $\left\lfloor \frac{n(n+1)}{2} \right\rfloor$ represents the greatest integer less than or equal to $\frac{n(n+1)}{2}$. Since $\frac{n(n+1)}{2}$ is already an integer, we can simply consider the integer value without needing the floor function.

Step 3: Add 1 to the Expression

Now, we add 1 to the value:

$\left\lfloor \frac{n(n+1)}{2} \right\rfloor + 1$

So we need to analyze the parity (even or odd nature) of $\frac{n(n+1)}{2}$, then add 1:

• If $\frac{n(n+1)}{2}$ is even, then $\frac{n(n+1)}{2} + 1$ is odd.
• If $\frac{n(n+1)}{2}$ is odd, then $\frac{n(n+1)}{2} + 1$ is even.

Step 4: Conditions Based on $n$

Now we focus on the parity of $\frac{n(n+1)}{2}$:

• $\frac{n(n+1)}{2}$ is even when $n(n+1)$ is divisible by 4.
• $\frac{n(n+1)}{2}$ is odd when $n(n+1)$ is 2 modulo 4.

Conclusion

• The expression $\left\lfloor \frac{n(n+1)}{2} \right\rfloor + 1$ is even when $\frac{n(n+1)}{2}$ is odd, i.e., when $n \equiv 1 \pmod{4}$.
• The expression is odd when $\frac{n(n+1)}{2}$ is even, i.e., when $n \equiv 0 \pmod{4}$.

Final Answer:

• The expression is odd when $n \equiv 0 \pmod{4}$.
• The expression is even when $n \equiv 1 \pmod{4}$.

Would you like further clarification or more examples?

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1. What is the general formula for the sum of the first $n$ integers?
2. How does modular arithmetic help determine the parity of a number?
3. What is the behavior of the expression when $n$ is even or odd?
4. Can the expression be simplified further for specific values of $n$?
5. How can this kind of problem be applied in other areas of mathematics?

Tip: Try testing the formula with small values of $n$ (e.g., $n = 0, 1, 2, 3$) to visually confirm the parity pattern.