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 \(n\).

Guest Oct 23, 2019

#1**+1 **

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 2^{n} (there's a really cool counting proof for this but you probabally don't need to know that)

so f(n)=log(2^{n})

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 \(\frac{f(n)}{\log_{10}2}=\boxed{n}\)

.power27 Oct 24, 2019