Galileo's ratios with the odd numbers

Text

Galileo observed that $\frac{1}{3}=\frac{1+3}{5+7}=\frac{1+3+5}{7+9+11}=\cdots$ Check the relation $\dfrac{1+3+5+7}{9+11+13+15}$ and prove in general that the sum of the first $k$ odd numbers, divided by the sum of the next $k$ odd numbers, always equals $\frac13$.

Solution

For $k=4$: numerator $1+3+5+7=16$, denominator $9+11+13+15=48$, and $\frac{16}{48}=\frac13$. In general the sum of the first $k$ odds is $1+3+\cdots+(2k-1)=k^2$. The sum of the first $2k$ odds is $(2k)^2=4k^2$, so the sum of the next $k$ odds is $4k^2-k^2=3k^2$. The ratio is $\frac{k^2}{3k^2} = \boxed{\frac{1}{3}},$ independent of $k$.