a Putnamexamlevel system of equations  Summary and Q&A
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.
Questions & Answers
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*...(2n1))/(246...(2n)). For odd terms, the formula is (135...(2n1))/(246...*(2n2)).
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*(2n2)(2n4)...2)/(135...(2n1)). This can be simplified to 2/(135...*(2n1)).
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.