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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 \(n\).

 Oct 23, 2019
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The rows of Pascal's triangle sum to powers of 2, so the elements of the nth row (the row with only one element in it being row 0) sum to 2n (there's a really cool counting proof for this but you probabally don't need to know that)

so f(n)=log(2n)

Put this into the fraction:

\(\frac{\log{2^{n}}}{\log{2}}\)

Pop that n out front from logarithm laws:

\(\frac{n\cdot\log{2}}{\log{2}}\)

The log(2)s cancel out and you are left with \(\frac{f(n)}{\log_{10}2}=\boxed{n}\)

.
 Oct 24, 2019

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