en. prove by induction ∑k = 1n k ( k + 1) = n ( n + 1) ( n + 2) 3. induction-calculator. Solution to Problem 3: Statement P (n) is defined by 1 3 + 2 3 + 3 3 + ... + n 3 = n 2 (n + 1) 2 / 4STEP 1: We first show that p (1) is true.Left Side = 1 3 = 1Right Side = 1 2 (1 + 1) 2 / 4 = 1 hence p (1) is true. Study it well! If this is your first visit to this page you may want to check out the help page. $prove\:by\:induction\:\sum_ {k=1}^nk\left (k+1\right)=\frac {n\left (n+1\right)\left (n+2\right)} {3}$. The above is a well explained and solid proof by mathematical induction. Mathematical Induction Solver This page was created to help you better understand mathematical induction. This tool can help you gain a better understanding of your hypothesis and can prove the hypothesis false. prove by induction ∑k = 1n k3 = n2 ( n + 1) 2 4. Since 2 = 1 × 2 and 1 + 2 = 3, k 2 + 3k + 2 = ( k + 1) × ( k + 2) Therefore, 2 + 4 + 6 + ... + 2k + 2 × ( k + 1) = ( k + 1) × ( k + 2) and the proof by mathematical induction is complete!