Understanding the Series
The series given is a combination of fractions and products that can be simplified to identify a pattern. Each term in the series contributes to the overall sum, and recognizing how each term is formed will help in solving it.
Breakdown of Terms
- The first term is straightforward:
1
- The second term involves division:
2 ÷ 6 = 1/3
- The third term can be computed step by step:
2 × 5 ÷ 6 × 12 = (10 ÷ 6) × 12 = 20/6 = 10/3
- The fourth term is more complex:
2 × 5 × 8 ÷ 6 × 12 × 18 = (80 ÷ 6) × 12 × 18 = 960/6 = 160
Identifying the Pattern
- Each term can be expressed in a factorial-like structure:
- The numerator increases multiplicatively (2, 2×5, 2×5×8, ...).
- The denominator incorporates the factorial of the terms involved with increasing complexity.
Summation of the Series
- This series can be recognized as a summation of a specific mathematical function, which may converge to a particular value as more terms are added.
- The pattern can often be related to exponential functions or series expansions.
Conclusion
To solve this series completely, one would typically look for a closed-form expression or convergence properties using mathematical tools such as the Binomial Theorem or calculus. Each term contributes to the overarching behavior of the series, and understanding this is key to mastering series summation techniques in mathematics.

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