Sequences & Series
Sequence and Series
star_batch_jee_advanced_2025
Grade 11
Question:
If the value of $\sum_{i=0}^{n} \sum_{j=0}^{n} \sum_{k=0}^{n} \frac{1}{3^{i+j+k}}$ ($i \neq j \neq k$) is equal to $\frac{m}{n}$, where $m, n$ are coprime natural numbers, then $m + n$ is equal to ____.
Step-by-Step Solution
Key Concept: Use inclusion-exclusion to account for distinct index constraints by subtracting degenerate cases from the full power sum.
The triple sum with condition $i \neq j \neq k$ is evaluated by subtracting overcounted cases. The full triple sum equals $\left(\sum \frac{1}{3^i}\right)^3 = \left(\frac{1}{1-1/3}\right)^3 = \left(\frac{3}{2}\right)^3 = \frac{27}{8}$. Subtracting the diagonal and pair terms gives $\frac{27}{8} - \frac{3}{2} \times 3 \times \frac{1}{1-1/9} + 2 \times \frac{1}{1-1/27} = \frac{81}{208}$.
Correct Answer: 289