What is the smallest n for a small inequality?

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SUMMARY

The discussion centers on determining the smallest integer n that satisfies the inequality lg {n\choose0.15n} + 0.15n \geq 112. Participants clarify that lg refers to the logarithm base 2 and emphasize that the logarithm applies solely to the binomial coefficient {n\choose0.15n}. A consensus emerges that numerical methods may provide the most effective solution for this problem.

PREREQUISITES
  • Understanding of logarithmic functions, specifically lg (log base 2).
  • Familiarity with binomial coefficients, denoted as {n\choose k}.
  • Basic knowledge of inequalities and how to manipulate them.
  • Experience with numerical methods for solving mathematical problems.
NEXT STEPS
  • Explore numerical methods for solving inequalities, focusing on techniques such as binary search.
  • Study the properties of binomial coefficients and their applications in combinatorial mathematics.
  • Learn about logarithmic identities and their implications in mathematical inequalities.
  • Investigate software tools or programming languages that can assist in numerical computations, such as Python's NumPy library.
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Mathematicians, students studying combinatorics, and anyone interested in solving inequalities using numerical methods.

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What is the smallest n such that

\lg {n\choose0.15n} + 0.15n \geq {112}
 
Last edited:
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Dragonfall said:
What is the smallest n such that

\lg {n\choose0.15n} + {n\choose0.15n} \geq {112}
What is lg?
 
Math_QED said:
What is lg?

Log base 2
 
Dragonfall said:
Log base 2
Does the log contain the sum of both combinations? (Then you should have added more brackets)
 
Math_QED said:
Does the log contain the sum of both combinations? (Then you should have added more brackets)

Sorry I formulated the inequality wrong. It's fixed now.

The log only contains the binomial.

I think it might be easier just to do this numerically...
 
You can try to do:

2^LH = 2^RH

But I think the best approach is a numerical method.
 

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