Final answer:
Using mathematical induction, we can prove that the equation 1³ + 2³ + 3³ + ... + n³ = [n(n+1)/2]² holds true for every positive integer n.
Step-by-step explanation:
We will use mathematical induction to prove the equation 1³ + 2³ + 3³ + ... + n³ = [n(n+1)/2]².
Step 1: Base case:
When n = 1, we have 1³ = [1(1+1)/2]² = 1, which is true.
Step 2: Inductive hypothesis: Assume that the equation holds true for some positive integer k, i.e., 1³ + 2³ + 3³ + ... + k³ = [k(k+1)/2]².
Step 3: Inductive step: We need to prove that the equation also holds true for k+1.
Adding (k+1)³ to both sides of the equation, we get:
1³ + 2³ + 3³ + ... + k³ + (k+1)³ = [k(k+1)/2]² + (k+1)³
Simplifying the right side, we have:
[k²(k+1)²/4] + [(k+1)³/1] = [(k³ + 3k² + 3k + 1)(k+1)²/4]
Expanding and simplifying further, we get:
[(k⁴ + 4k³ + 6k² + 4k + 1)/4][(k+1)²] = [(k+1)⁴/4]
Therefore, the equation holds true for k+1.
Step 4: Conclusion: By the principle of mathematical induction, the equation is true for every positive integer n.