If 1/(√1+√2) + 1/(√2+√3) + ... + 1/(√99+√100) = m and 1/(1·2) + 1/(2·3) + ... + 1/(99·100) = n, then

Q: Wait, do I really need to write out all 99 terms to see the cancellation?

Absolutely not! The beauty of a telescoping sum is that you only need to write the first two or three terms and the last term to identify the pattern. Once you see the 'cancellation chain', you can confidently jump to the result.

Q: Why did we use partial fractions for the second series but rationalization for the first?

Great observation! We use rationalization when we have irrational terms (roots) to make the denominator rational. We use partial fractions when we have a product of factors in the denominator to break it into a difference. Both techniques aim for the same goal: creating a telescoping structure.

Q: Is there a shortcut to finding the line equation?

While you could test the options, the most robust way is to find the exact coordinates $(m, n)$ first. Once you have $(9, 99/100)$, testing the options is just a quick arithmetic check. Never guess when you can calculate!

Question

If $\frac{1}{\sqrt{1}+\sqrt{2}} + \frac{1}{\sqrt{2}+\sqrt{3}} + \dots + \frac{1}{\sqrt{99}+\sqrt{100}} = m$ and $\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \dots + \frac{1}{99 \cdot 100} = n$, then the point $(m, n)$ lies on the line

Options:

(A) $11(x-1) - 100(y-2) = 0$

(B) $11x - 100y = 0$

(C) $11(x-2) - 100(y-1) = 0$

(D) $11(x-1) - 100y = 0$

Sigma Percentile · Accepted Answer

### The Art of Telescoping: Unlocking Infinite Series

Imagine you are standing before a massive, intimidating wall of numbers. At first glance, the series

$$m = \sum_{r=1}^{99} \frac{1}{\sqrt{r}+\sqrt{r+1}}$$

and

$$n = \sum_{r=1}^{99} \frac{1}{r(r+1)}$$

look like they might take hours to compute. But in the world of JEE Advanced, we don't brute force; we look for the hidden symmetry.

### Phase 1: The Irrational Challenge

Let us focus on $m$. The denominator $\sqrt{r} + \sqrt{r+1}$ is the enemy of simplicity. It is irrational, and it is messy.

But we have a weapon: **rationalization**. By multiplying the numerator and denominator by the conjugate $\sqrt{r+1} - \sqrt{r}$, we transform the denominator into:

$$(\sqrt{r+1}+\sqrt{r})(\sqrt{r+1}-\sqrt{r}) = (r+1) - r = 1$$

Suddenly, the general term $T_r$ becomes $\sqrt{r+1} - \sqrt{r}$. This is the breakthrough!

### Phase 2: The Telescoping Magic

Now, watch what happens when we sum these terms from $r=1$ to $99$:

$$m = (\sqrt{2}-\sqrt{1}) + (\sqrt{3}-\sqrt{2}) + (\sqrt{4}-\sqrt{3}) + \dots + (\sqrt{100}-\sqrt{99})$$

Look closely at the pattern. The positive $\sqrt{2}$ cancels the negative $\sqrt{2}$, and the positive $\sqrt{3}$ cancels the negative $\sqrt{3}$.

This is a **telescoping sum**—it collapses like a pirate's telescope, leaving only the very last term and the very first term. Thus:

$$m = \sqrt{100} - \sqrt{1} = 10 - 1 = 9$$

### Phase 3: The Rational Decomposition

With $m$ conquered, we turn to $n = \sum_{r=1}^{99} \frac{1}{r(r+1)}$. Here, we don't have roots, but we have a product in the denominator.

The method of **partial fractions** is our key. We can rewrite the general term as:

$$T_r = \frac{(r+1)-r}{r(r+1)} = \frac{1}{r} - \frac{1}{r+1}$$

Again, we see the telescoping structure emerge. Expanding this, we get:

$$\left(1 - \frac{1}{2}\right) + \left(\frac{1}{2} - \frac{1}{3}\right) + \dots + \left(\frac{1}{99} - \frac{1}{100}\right)$$

The middle terms vanish, leaving us with:

$$n = 1 - \frac{1}{100} = \frac{99}{100}$$

### Phase 4: The Geometric Conclusion

We have our coordinates: $(m, n) = (9, \frac{99}{100})$. The final step is to find which line passes through this point.

Testing the equation $11x - 100y = 0$, we substitute $x=9$ and $y=\frac{99}{100}$:

$$11(9) - 100\left(\frac{99}{100}\right) = 99 - 99 = 0$$

It fits perfectly! You have just navigated a complex series problem by identifying the underlying structure. Remember, in physics and math, the most complex problems are often just simple patterns waiting to be revealed.

Animated Solution for Mathematics - Sequence and Series: If $\frac{1}{1 + 2} + \frac{1}{2 + 3} + \dots + \frac{1}{99 + 100} = m$ and $\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \dots + \frac{1}{99 \cdot 100} = n$ , then the point $(m, n)$ lies on the line

Select Answer:

Visualized Solution

Analyze the Structure of $m$

Rationalize the General Term of $m$

Expand and Telescope $m$

Calculate Final Value of $m$

Analyze the Structure of $n$

Partial Fraction Decomposition for $n$

Expand and Telescope $n$

Calculate Final Value of $n$

Identify the Point

Verify the Line Equation

The Sigma Insight: Sum of Special Series

The Art of Telescoping

Phase 1

Phase 2

Phase 3

Phase 4

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Select Answer:

Visualized Solution

Analyze the Structure of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } m

Rationalize the General Term of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } m

Expand and Telescope .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } m

Calculate Final Value of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } m

Analyze the Structure of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } n

Partial Fraction Decomposition for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } n

Expand and Telescope .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } n

Calculate Final Value of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } n