The sum from n=1 to 10 of (528 / (n(n+1)(n+2))) is equal to:

The correct answer is Option 2. Full mathematical solution steps and hints are accessible in the interactive StemCET workspace.

Maths Q5 – JEE-MAIN 2026

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Home Maths Sequences and SeriesQuestion #396

\sum_{n = 1}^{10} (\frac{528}{n (n + 1) (n + 2)}) is equal to:

Clue 1 of 3: Recognize the structure

The general term of the series involves a product of consecutive integers in the denominator, suggesting the use of partial fraction decomposition.

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Step 1: Decompose the General Term using Partial Fractions✦ Active

Let the general term be $T_{n} = \frac{1}{n (n + 1) (n + 2)}$ . We can decompose this using partial fractions. A common approach for such products is to notice that:

\frac{1}{n (n + 1) (n + 2)} = \frac{1}{2} [\frac{(n + 2) - n}{n (n + 1) (n + 2)}] = \frac{1}{2} [\frac{1}{n (n + 1)} - \frac{1}{(n + 1) (n + 2)}]

Further, we know that $\frac{1}{k (k + 1)} = \frac{1}{k} - \frac{1}{k + 1}$ . Applying this:

T_{n} = \frac{1}{2} [(\frac{1}{n} - \frac{1}{n + 1}) - (\frac{1}{n + 1} - \frac{1}{n + 2})]

Step 2: Identify the Telescoping Sum Pattern○ Expand

Let $f (n) = \frac{1}{n} - \frac{1}{n + 1}$ . Then the general term $T_{n}$ can be written as:

T_{n} = \frac{1}{2} [f (n) - f (n + 1)]

The sum is given by $\sum_{n = 1}^{10} 528 \cdot T_{n}$ . Substituting $T_{n}$ :

S = 528 \sum_{n = 1}^{10} \frac{1}{2} [f (n) - f (n + 1)] = 264 \sum_{n = 1}^{10} [f (n) - f (n + 1)]

This is a telescoping sum. When expanded, most terms cancel out:

S = 264 [(f (1) - f (2)) + (f (2) - f (3)) + \dots + (f (10) - f (11))]

The sum simplifies to:

S = 264 [f (1) - f (11)]

💡 Teacher's Secret Hint

Ensure the partial fraction decomposition is correct to reveal the telescoping nature.

Step 3: Evaluate the Sum○ Expand

Now, we calculate $f (1)$ and $f (11)$ :

f (1) = \frac{1}{1} - \frac{1}{1 + 1} = 1 - \frac{1}{2} = \frac{1}{2}

f (11) = \frac{1}{11} - \frac{1}{11 + 1} = \frac{1}{11} - \frac{1}{12} = \frac{12 - 11}{11 \times 12} = \frac{1}{132}

Substitute these values back into the simplified sum expression:

S = 264 [\frac{1}{2} - \frac{1}{132}]

Find a common denominator for the terms inside the bracket:

S = 264 [\frac{66}{132} - \frac{1}{132}] = 264 [\frac{65}{132}]

Perform the multiplication:

S = \frac{264 \times 65}{132}

Since $264 = 2 \times 132$ :

S = 2 \times 65 = 130

💡 Teacher's Secret Hint

Be careful with arithmetic, especially when combining fractions.

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Maths Question 5 – JEE-MAIN 2026

Clue 1 of 3: Recognize the structure

🧠 Full Solution Path