Problem 53 - Precalculus with Limits

Question

In Exercises 49 - 58, find the sum using the formulas for the sums of powers of integers. \( \sum_{n=1}^{5}n^4 \)

Step-by-Step Solution

Answer

The sum of the fourth powers of integers from 1 to 5 is 1485.

Step 1: Identify the Formula

Identify the formula for the sum of fourth powers, which is \\[ \frac{n(n+1)(2n+1)(3n^2+3n-1)}{30} \\]. This formula enables us to find sum of fourth powers of integers.

Step 2: Apply the Formula

Insert \(n=5\) into our formula. This will give us: \\[ \frac{(5)(5+1)(2*5+1)(3*5^2+3*5-1)}{30} \\]

Step 3: Simplify the Equation

Simplify the equation by doing the calculations step by step: \\[ \frac{5*6*11*(135) }{30} = \frac{5*6*11*3*45}{30} = \frac{5*3*11*90 }{10}=5*3*11*9 = 1485 \\]

Key Concepts

Sum of Fourth Powers

Understanding the sum of fourth powers involves dealing with a series where each term is the fourth power of an integer. Specifically, the sum of the n first fourth powers can be represented as:
\[ \sum_{k=1}^{n}k^4 \]
This notation expresses the sum of the series where 'k' is raised to the fourth power and 'k' varies from 1 to 'n'. The formula for this sum is:
\[ \sum_{k=1}^{n}k^4 = \frac{n(n+1)(2n+1)(3n^2+3n-1)}{30} \]
In this specific case, to find \( \sum_{n=1}^{5}n^4 \), one must substitute 'n' with 5 in the formula. This process is crucial as it simplifies the approach to calculating the series sum without manually adding each number's fourth power, which is time-consuming and prone to errors. The utilization of this formula allows for a logical and streamlined method to resolve the problem.

Mathematical Induction

Mathematical induction is a powerful proof technique used widely in mathematics, especially in series and sequences. To connect with our current context, once a formula is known for a sum of powers, like the sum of fourth powers, we often use induction to prove that the formula works for all positive integers.

The principle of mathematical induction works in two steps: First, we prove that the formula or statement is true for the first integer in our series, often when 'n=1'. Second, we assume that the formula is true for some integer 'k', and then prove that it must be true for the next integer 'k+1'. This 'domino effect' establishes the truth of the statement for all integers greater than or equal to our starting point.
In practice, for the sum of fourth powers, after finding a formula, induction could be used to demonstrate that the formula is indeed valid for calculating the sum for any number 'n'.

Series and Sequences

When we refer to series and sequences, we are talking about an ordered list of numbers following a specific rule. A sequence is simply the list of numbers in order, while the series is the sum of the terms of the sequence. An example of a sequence is the list of squares: \(1^2, 2^2, 3^2, \ldots\), and its corresponding series (where we sum the terms) would be \(1^2 + 2^2 + 3^2 + \ldots\).

In the context of sums of powers, like the sum of fourth powers, the sequence consists of each integer raised to the fourth power individually, and the series is the sum of these terms. This knowledge is fundamental, as by understanding it, one can start to see patterns and relationships between the terms. Recognizing these patterns is essential in deriving the formulas like the one used for the sum of fourth powers. Furthermore, familiarizing oneself with the behavior of series can not only aid in solving problems but also in predicting the properties of similar mathematical expressions.