Problem 24 - University Calculus: Early Transcendentals

Question

Evaluate the sums. $$\sum_{k=1}^{6}\left(k^{2}-5\right)$$

Step-by-Step Solution

Answer

The sum is 61.

Step 1: Understand the Formula

The problem asks us to evaluate the sum of the expression \(k^2 - 5\) from \(k = 1\) to \(k = 6\). This means we compute \((1^2 - 5) + (2^2 - 5) + (3^2 - 5) + (4^2 - 5) + (5^2 - 5) + (6^2 - 5)\).

Step 2: Calculate Each Term

Calculate each term in the series: 1. For \(k = 1\), \(1^2 - 5 = 1 - 5 = -4\).2. For \(k = 2\), \(2^2 - 5 = 4 - 5 = -1\).3. For \(k = 3\), \(3^2 - 5 = 9 - 5 = 4\).4. For \(k = 4\), \(4^2 - 5 = 16 - 5 = 11\).5. For \(k = 5\), \(5^2 - 5 = 25 - 5 = 20\).6. For \(k = 6\), \(6^2 - 5 = 36 - 5 = 31\).

Step 3: Sum the Results

Add all the values obtained in Step 2: \(-4 + (-1) + 4 + 11 + 20 + 31 = 61\).

Step 4: Conclusion

Thus, the sum of \(\sum_{k=1}^{6} (k^2 - 5)\) is 61.

Key Concepts

Series

A series is a way to add up a sequence of numbers.
It involves taking several terms and adding them together into one sum.
This is particularly useful for long or infinite sequences. In the context of our exercise, we are dealing with a finite series, where we add the results for values of

• \( k^2 - 5 \)
from \( k = 1 \) to \( k = 6 \).

Series can be categorized into different types, such as arithmetic or geometric series, though this particular series does not fit neatly into those categories, as it involves squaring each term \( k \). Understanding series helps simplify complex calculations by managing numerous operations into a single, coherent process.

Arithmetic operations

Arithmetic operations are basic mathematical procedures such as addition, subtraction, multiplication, and division. These operations are the building blocks for solving any mathematical problem, including evaluating series.
In the given problem, we use:

• Subtraction to simplify each term, \(k^2 - 5\), and
• Addition to sum all the results together.

Performing arithmetic correctly ensures accuracy in the evaluation process. Properly executing these operations allows us to solve the series step by step and reach the final result confidently.

Algebra

Algebra involves using symbols and letters to represent numbers and express mathematical relationships. It is a broad topic that includes equations, functions, and expressions.
In our exercise, the expression \(k^2 - 5\) is an algebraic expression formed by combining a square term, \(k^2\), and a constant \(-5\). We use algebraic manipulation to simplify these expressions for each value of \(k\), from \(1\) to \(6\). By understanding these relationships, solving the series becomes straightforward. Algebra thus serves as a fundamental tool for expressing complex problems in a manageable format.

Mathematical notation

Mathematical notation consists of symbols and signs used to represent mathematical concepts and operations clearly and concisely. It allows mathematicians to communicate complex ideas succinctly.
In this exercise:

• The summation symbol, \(\sum\), denotes that we need to sum all terms.
• The variable \(k\) is used as an index that changes from \(1\) to \(6\).
• The expression \(k^2 - 5\) defines the sequence we are adding.

Using correct notation not only helps us understand the problem but also maintains precision throughout the calculation process. Mastery of such notation is crucial for anyone looking to excel in mathematics.