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Let \(f(n)\) be the base-10 logarithm of the sum of the elements of the \(n\)th row in Pascal's triangle. Express \(\frac{f(n)}{\log_{10} 2}\) in terms of \(n\). Recall that Pascal's triangle begins

1   -> n=0

1     1   -> n=0

1     2     1   -> n=0

1     3     3     1   -> n=0

1     4     6     4     1   -> n=0

...                           ...               

 Feb 24, 2019
 #1
avatar+4404 
+2

\(\text{the }n \text{th row of Pascal's triangle is given by}\\ \left\{\dbinom{n}{k}:k=0,n\right\}\)

 

\(f(n) = \log_{10}\left(\sum \limits_{k=0}^n \dbinom{n}{k}\right) = \log_{10}\left(2^n \right)= n \log_{10}(2)\\ \dfrac{f(n)}{\log_{10}(2)} = n\)

.
 Feb 25, 2019

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