If [x] be the greatest integer less than or equal to x, then Σ(n=8 to 100) [(-1)^n n / 2] is equal t

Q: Wait, why does $[-4.5]$ become $-5$ and not $-4$? Isn't $-4$ greater than $-4.5$?

This is the classic 'GIF Trap'! You are absolutely correct that $-4 > -4.5$. However, the definition of the Greatest Integer Function $[x]$ is the largest integer that is *less than or equal to* $x$. On the number line, 'less than' always points to the left. Since $-5$ is to the left of $-4.5$, it is the largest integer satisfying the condition. Always visualize the number line to avoid this mistake!

Q: Can I just use the property $[x+I] = [x] + I$ to solve this?

You certainly can, but you must be extremely careful with the negative sign. For odd $n$, you have $[-n/2]$. If you write this as $-[n/2]$, you will get the wrong answer. The correct property is $[-x] = -[x] - 1$ if $x$ is not an integer. Using this property will lead you to the same result, but it requires more algebraic rigor than simply evaluating the first few terms to find the pattern.

Q: Is there a formula for this sum, or do I have to write it all out?

While you could derive a general formula for the sum, in JEE Advanced, the 'write it out' method is often the safest and fastest way to spot a telescoping series. By writing out the first few terms, you reveal the symmetry of the problem. Pattern recognition is a skill that is just as important as algebraic manipulation!

Question

If $[x]$ be the greatest integer less than or equal to $x$, then $\sum_{n=8}^{100} \left[ \frac{(-1)^n n}{2} \right]$ is equal to:

Options:

(A) 0

(B) 4

(C) -2

(D) 2

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, future engineers. Today, we are going to peel back the layers of a problem that looks intimidating at first glance but hides a beautiful, rhythmic symmetry. We are tasked with evaluating the sum:

$$S = \sum_{n=8}^{100} \left[ \frac{(-1)^n n}{2} \right]$$

At first, the Greatest Integer Function (GIF) and the alternating sign $(-1)^n$ might seem like they are trying to confuse you. But let us take a breath and look at the soul of this expression.

### Phase 1: The Even-Odd Split

The term $(-1)^n$ is the heartbeat of this series. It dictates the behavior of every single term.

When $n$ is even, $(-1)^n$ becomes $1$, and our expression simplifies to $\left[ \frac{n}{2} \right]$. When $n$ is odd, $(-1)^n$ becomes $-1$, and we are left with $\left[ \frac{-n}{2} \right]$.

This even-odd split is our roadmap. Let us start by calculating the first few terms to see what happens.

For $n=8$ (even): We have $\left[ \frac{8}{2} \right] = [4] = 4$.

But now, look at $n=9$ (odd). We have $\left[ \frac{-9}{2} \right] = [-4.5]$.

Here is where the trap lies. Many students instinctively want to round $-4.5$ to $-4$. But remember, the Greatest Integer Function is a floor function. It demands the largest integer less than or equal to the input.

On the number line, we must look to the left. The first integer to the left of $-4.5$ is $-5$. So, $[-4.5] = -5$.

### Phase 2: The Telescoping Magic

Let us continue this journey. For $n=10$ (even), we get $\left[ \frac{10}{2} \right] = [5] = 5$. For $n=11$ (odd), we get $\left[ \frac{-11}{2} \right] = [-5.5] = -6$.

Do you see it? The sequence of terms is $4, -5, 5, -6, 6, -7, 7, \dots$.

Look closely at the pairs. We have a $-5$ followed by a $5$. We have a $-6$ followed by a $6$. This is a classic telescoping series!

Every odd term $n$ produces a negative integer, and the very next even term $n+1$ produces the exact same integer with a positive sign. They are destined to cancel each other out.

The sum is essentially:

$$S = 4 + (-5 + 5) + (-6 + 6) + \dots + (-50 + 50)$$

### Phase 3: The Final Cancellation

To be absolutely sure, let us look at the end of our series. The last term is $n=100$.

Since $100$ is even, we have $\left[ \frac{100}{2} \right] = 50$. The term before it, $n=99$, is odd, so we have $\left[ \frac{-99}{2} \right] = [-49.5] = -50$.

The pair $(-50 + 50)$ sums to zero, just like all the pairs before it. The only term left without a partner is our very first term, $4$.

### Conclusion: The Elegance of Simplicity

When we sum everything up, the entire series collapses into a single value: **$4$**.

It is a powerful reminder that in mathematics, complexity is often just a mask for underlying simplicity. When you encounter these problems in the exam hall, do not panic.

Write out the first few terms, visualize the number line, and look for the pattern. You have the tools; now go forth and solve with confidence!

Animated Solution for Mathematics - Functions: If $[x]$ be the greatest integer less than or equal to $x$ , then $\sum_{n = 8}^{100} [\frac{( - 1 ) ^{n} n}{2}]$ is equal to:

Select Answer:

Visualized Solution

Understanding the Summation

Analyzing Even and Odd $n$

Evaluating the First Term ( $n = 8$ )

Evaluating the Second Term ( $n = 9$ )

The Greatest Integer Trap

Evaluating $n = 10$ and $n = 11$

Generalizing the Sequence

Evaluating the Final Terms

Grouping and Canceling Terms

Final Calculation

The Sigma Insight: Classification of Functions

Analyzing the Setup

Phase 1

Phase 2

Phase 3

Conclusion

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Animated Solution for Mathematics - Functions: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If [x] be the greatest integer less than or equal to x, then ∑n=8100​[2(−1)nn​] is equal to: