Prove that the sum of odd numbers is a perfect square

by RoRi

July 5, 2015

Prove $\displaystyle{\sum_{k=1}^n (2k-1) = n^2}.$

Proof. We write, $2k - 1 = k^2 - (k^2 - 2k +1) = k^2 - (k-1)^2$ which gives us a telescoping sum,

$\begin{align*} \sum_{k=1}^n (2k-1) &= \sum_{k=1}^n (k^2 - (k-1)^2) \\ &= n^2 - 0 & (\text{Telescoping property}). \\ &= n^2. \qquad \blacksquare \end{align*}$

Stumbling Robot

Prove that the sum of odd numbers is a perfect square

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