Final answer:
The proof by induction involves confirming the base case and then showing that if an equation holds for an arbitrary natural number k, it also holds for k+1 by using algebraic manipulation.
Step-by-step explanation:
The question is asking us to prove by induction a formula involving a product of sequential terms and to show it equals a given expression. To address this, we first need to understand the principles of mathematical induction and how they can be applied to demonstrate that a given property holds true for all natural numbers.
Base Case
Start with n=1 to show that the base case holds:
When n=1, the left side of the equation becomes 1(1+1), which is simply 2.
The right side of the equation, according to the formula, is \(\frac{1}{3} n(n+1)(n+2)\), which when n=1, also calculates to 2.
Since both sides are equal when n=1, the base case holds.
Inductive Step
Assume the statement is true for some arbitrary natural number k. Then we must show that it also holds for k+1. This step will involve algebraic manipulation where we will use the inductive hypothesis to simplify the expression for k+1 terms and then show that it is equal to the right side of the equation which is \(\frac{1}{3} (k+1)(k+2)(k+3)\).
The detailed algebraic steps will ensure that every part of the induction process is covered, eventually leading to the conclusion that if the statement holds for k, it indeed holds for k+1, thus completing the proof by induction.