Proving Mathematical Induction in the Sum of Sequence Formulas: 1223...nn1 [nn1 n2]/3

Introduction:

The concept of mathematical induction is a powerful tool in mathematics, used to prove statements about all positive integers. In this article, we will demonstrate the application of mathematical induction to prove the formula:

1223...nn1 [nn1 n2]/3

This article is designed to help students and educators understand the intricacies of mathematical induction, with a step-by-step guide to proving the given formula.

Understanding Mathematical Induction

Mathematical induction involves two steps:

tBase Case: Proving the formula holds for the smallest value, usually n1. tInductive Step: Assuming the formula holds for some arbitrary value t, prove it holds for the next value t 1.

Base Case: n1

First, let's establish the base case. For n1:

12 2

On the other side, [1 2]/3 2/3 * 3 2.

Both sides yield 2, verifying the base case.

Inductive Step: From nt to nt 1

Now we assume the formula is true for some value t:

1223...t(t 1) [t(t 1)(t 2)]/3

Next, we add the next term to the sequence. We need to prove that:

(t 1) [(t 1)(t 2)(t 3)]/3

Starting from the sum up to t and adding the next term:

[t(t 1)(t 2)]/3 (t 1)(t 2)

Factor out common terms:

[t(t 1)(t 2)]/3 (t 1)(t 2) (t 1)(t 2)[t/3 1] (t 1)(t 2)[(t 3)/3]

Which simplifies to:

[(t 1)(t 2)(t 3)]/3

This confirms the inductive step, thus the formula holds for any n by the principle of mathematical induction.

Step-by-Step Proof

Let's break down the proof step-by-step:

tBase Case: t ttWe verify the formula holds for n1 by showing that both sides are equal to 2. t tInductive Step: t ttAssume the formula holds for nt. tt tttShow that if the formula holds for nt, then it also holds for nt 1 by algebraic manipulation. tt ttThe proof involves the following steps: tt tttWrite the formula up to t. tttAdd the next term, factor out common terms. tttSimplify to the desired form. tt ttThis shows the formula is true for all positive integers n. t

With the base case verified and the inductive step proven, we have demonstrated the formula using mathematical induction.

Conclusion

The proof by mathematical induction confirms that:

1223...nn1 [nn1 n2]/3

for all positive integers n. This example illustrates the power and rigor of mathematical induction in proving sequence formulas.

For further practice, consider applying this technique to other sequence formulas, enhancing your understanding and proficiency in mathematical induction.