Answer:
Step-by-step explanation:
To prove the given statement using mathematical induction, we follow two main steps: the base case and the inductive step.
Base Case: We start by verifying the statement for ( n = 1 ): [ \sum_{k=1}^{1} k^3 = 1^3 = 1 ] [ \frac{12(1+1)2}{4} = \frac{1 \cdot 2^2}{4} = 1 ] Since both sides equal 1, the base case holds true.
Inductive Step: Assume the statement is true for some integer ( n ), which is our inductive hypothesis: [ \sum_{k=1}^{n} k^3 = \frac{n2(n+1)2}{4} ]
Now we need to prove the statement for ( n + 1 ): [ \sum_{k=1}^{n+1} k^3 = \sum_{k=1}^{n} k^3 + (n+1)^3 ] Using the inductive hypothesis, we replace the sum up to ( n ): [ \frac{n2(n+1)2}{4} + (n+1)^3 ]
Simplify and factor to show that it equals to ( \frac{(n+1)2(n+2)2}{4} ), thus completing the inductive step.
Conclude that, by the principle of mathematical induction, the statement is true for all integers ( n \geq 1 ).