# a Putnam-exam-level system of equations | Summary and Q&A

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May 10, 2020
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a Putnam-exam-level system of equations

## TL;DR

Understanding the pattern and formula for finding the terms in an infinite system of equations and exploring the limit as n approaches infinity leads to the conclusion that a1 is equal to plus or minus the square root of 2/pi.

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### Q: How can we find the terms in an infinite system of equations?

By making observations and noticing patterns, we can derive a general formula. For even terms, the formula is (357*...(2n-1))/(246...(2n)). For odd terms, the formula is (135...(2n-1))/(246...*(2n-2)).

### Q: How does the limit as n approaches infinity relate to the terms in the equation?

The limit helps determine the behavior of the terms as n gets larger. In this case, the limit is used to analyze the ratio of a2n to a2n+1. The limit is found to be 1, indicating that the terms approach a certain value as n becomes infinitely large.

### Q: What is the formula for finding a2n/a2n+1?

The formula for finding a2n/a2n+1 is (2n*(2n-2)(2n-4)...2)/(135...(2n-1)). This can be simplified to 2/(135...*(2n-1)).

### Q: What is the value of a1 in the infinite system of equations?

The value of a1 is determined by taking the reciprocal of a2n/a2n+1 and finding its square root. This results in a1 being plus or minus the square root of 2/pi.

## Summary & Key Takeaways

• The content discusses how to find the terms in an infinite system of equations by observing patterns and using a general formula.

• The concept of limits is introduced, specifically the limit as n approaches infinity, and its relationship to the terms in the equation.

• The content demonstrates how the limit can be used to determine the value of a1, which is found to be plus or minus the square root of 2/pi.