Understanding and Proving Mathematical Induction for a Specific Sequence
This article explains the concept of mathematical induction and how to apply it to prove a specific sequence formula. The formula in question is: n × (n 1) × (n 2) / 3 Σ (i × (i 1)) for all positive integers n.
Introduction to Mathematical Induction
Mathematical induction is a powerful method used to prove statements about an infinite sequence of cases. It consists of two steps: the base case and the inductive step. The base case establishes the truth of the statement for the first instance. The inductive step assumes the statement is true for some arbitrary case, and then proves that it must also be true for the next case.
Correct Notation and Understanding the Sequence
The correct notation for the sequence is: n × (n 1) × (n 2) / 3. You may find confusion because commas are sometimes used to indicate multiplication, but this is not standard practice. Generally, the cross (#8901;) or dotted dot (#8905;) is used. For instance, 2 × 3 × 4 can be written as 2 × 3 × 4 or 2 · 3 · 4.
Base Case
The base case for this sequence is the smallest positive integer, which is 1. We need to show that the formula holds for n 1.
Substitute n 1 into the formula: 1 × 2 × 3 / 3 2
Calculate the left side: 1 × 2 × 3 / 3 6 / 3 2
Since both sides are equal, the base case is satisfied.
Inductive Step
For the inductive step, assume the formula is true for some positive integer k. That is:
k × (k 1) × (k 2) / 3 Σ (i × (i 1)) from i1 to k
We need to prove the formula is true for k 1.
Substitute n k 1 into the formula: (k 1) × (k 2) × (k 3) / 3
From the inductive hypothesis, we have: Σ (i × (i 1)) from i1 to k k × (k 1) × (k 2) / 3
We need to add the next term (k 1) × (k 2) to both sides:
(k × (k 1) × (k 2) / 3) (k 1) × (k 2) (k 1) × (k 2) × (k 3) / 3
Rearrange the left side: (k 1) × (k 2) [(k/3) 1] (k 1) × (k 2) [(k 3)/3]
Simplify the left side: (k 1) × (k 2) [(k 3)/3] (k 1) × (k 2) × (k 3) / 3
This matches the right side of the formula, proving the inductive step.
Conclusion
By proving the base case and the inductive step, we have shown that the formula n × (n 1) × (n 2) / 3 Σ (i × (i 1)) from i1 to n is true for all positive integers n. This is the essence of mathematical induction.
Further Resources
For further understanding, you may want to review your textbook's examples on mathematical induction. Websites like Math is Fun and Better Explained provide excellent explanations and examples.