Problem 21 - Precalculus with Limits

Question

In Exercises 11 - 24, use mathematical induction to prove the formula for every positive integer \( n \). \( \sum_{i=1}^{n}i^5 = \dfrac{n^2(n + 1)^2(2n^2 + 2n - 1)}{12} \)

Step-by-Step Solution

Answer

The formula \( \sum_{i=1}^{n}i^5 = \dfrac{n^2(n + 1)^2(2n^2 + 2n - 1)}{12} \) is proved for every positive integer \( n \) by using the principle of mathematical induction; by proving it for the base case when \( n = 1 \) and showing that, if it holds for \( n \), then it holds for \( n+1 \).

Step 1: Base Case

First, show that the formula holds for \( n=1 \). This is computed as \( \sum_{i=1}^{1}i^5 = 1^5 = 1 \) and checking the right-hand side of the formula \( \dfrac{1^2(1 + 1)^2(2(1)^2 + 2(1) - 1)}{12} = 1 \), which are equal.

Step 2: Inductive Step - Assume true for n

Next, assume that the formula holds for some arbitrary positive integer \( n \), i.e., \( \sum_{i=1}^{n}i^5 = \dfrac{n^2(n + 1)^2(2n^2 + 2n - 1)}{12} \). This is known as the inductive hypothesis.

Step 3: Inductive Step - Prove true for n+1

Now, one must prove that the formula holds for \( n+1 \) assuming it holds for \( n \). Thus, calculate the left-hand side for \( n+1 \), \( \sum_{i=1}^{n+1}i^5 \). This can be written as \( \sum_{i=1}^{n}i^5 + (n+1)^5 \), based on the definition of the sum. By substituting the inductive hypothesis into the equation, one obtains that \( \sum_{i=1}^{n+1}i^5 = \dfrac{n^2(n + 1)^2(2n^2 + 2n - 1)}{12} + (n+1)^5 \). Now, show that this statement equals the right-hand side of the formula for \( n+1 \), \( \dfrac{(n+1)^2(n + 2)^2(2(n+1)^2 + 2(n+1) - 1)}{12} \). The key here is rewritting the expression in an algebraically convenient form in order to match the form of the induction hypothesis.

Key Concepts

Proof by Induction

Mathematical induction is a powerful technique used to prove statements for all positive integers. It involves two main steps: the base case and the inductive step.
- **Base Case:** You start by proving that the statement is true for the initial value, typically when the positive integer, say \( n = 1 \). This establishes the truth of the proposition for the first step.- **Inductive Step:** Next, you assume the statement holds for some arbitrary integer \( n \). This assumption is called the inductive hypothesis. Then, you prove that if the statement holds for \( n \), it must also hold for \( n+1 \).
If both steps are successfully carried out, it implies that the formula is valid for all positive integers, demonstrating that your proposition follows logically from one integer to the next. This creates a chain reaction proving the whole sequence of numbers.

Sum of Powers

The sum of powers refers to summing up terms where each term is raised to a power, such as their squares, cubes, or higher powers like fifth powers. For instance, the sum \( \sum_{i=1}^{n}i^5 \) involves fifth powers of integers up to \( n \).
- **Meaning and Application:** This concept is crucial in algebra and number theory, providing insights into series and sequences.
- **Complexity:** Calculating the sum of these powers for any arbitrary value, especially when direct addition is not feasible, requires using established formulae, like the one in the original exercise.
These formulas provide a way to quickly compute the sum without manually adding each term, crucial for larger values of \( n \). These methods simplify calculations significantly.

Positive Integers

Positive integers are all the whole numbers greater than zero, i.e., \( 1, 2, 3, \dots \) and so on. These numbers form the backbone of number theory and are critical in understanding many mathematical concepts.
- **Properties:** These numbers do not include zero or any negative numbers. Each positive integer directly follows the other, contributing to their foundational role in sequences and series.
- **Use in Induction:** When proving a formula for all positive integers using induction, the initial proof for \( n=1 \) serves as the base to show that if it is valid for one, it should be valid for all that follow.
Positive integers are integral to creating systems such as counting and natural ordering, making them indispensable in mathematics.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying algebraic expressions to make them easier to handle or compare. In the context of proving summation formulas, manipulation is crucial to align both sides of an equation as part of the inductive step.
- **Techniques:** This includes factoring expressions, expanding terms, simplifying fractions, and breaking down complex parts into simpler ones. For example, adding \( (n+1)^5 \) to an already complex formula requires skill to maintain balance.- **Importance:** Through manipulation, one can transform an expression into a necessary format to prove identities or reach a logical conclusion.
Thus, algebraic manipulation is a key skill when working through problems in algebra, enabling solutions that meet given conditions or equivalences.