Problem 94 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the sum. $$\sum_{k=1}^{5} 4$$

Step-by-Step Solution

Answer

The sum of the sequence is 20.

Step 1: Understand The Sigma Notation and Terms

The given expression is in sigma notation, which is a convenient and compact way to express the sum of a sequence. Each term of the sequence here is constant, valued at 4.

Step 2: Calculate The Sum

Since all terms are equal to 4 - and there are five terms due to the range of \( k \) from 1 to 5 - the total sum can be found by multiplying 4 by 5.

Step 3: Summation Result

The result of \( 4 \times 5 = 20 \). Therefore, the total sum of the sequence is 20.

Key Concepts

Summation

Summation is a fundamental concept in mathematics that involves adding up a sequence of numbers. The notation for summation is the Greek letter sigma (\(\Sigma\)), which stands for the sum. When you see a sigma symbol, it's instructing you to sum the terms that follow a specific rule, from a starting point to an end point.

For example, in the expression \(\sum_{k=1}^{5} 4\), the sum requires adding the number 4 for each integer value of \(k\) from 1 to 5. This means you are adding the constant value 4, five times. Since the value does not change and is repeated a set number of times, it's easy to calculate the total sum by multiplying 4 by the number of terms, which is 5. As a result, \(4 \times 5 = 20\).

Sequence

A sequence is an ordered list of numbers following a specific pattern or rule. Each number in this list is called a term. Sequences are foundational in mathematics, as they allow you to describe and investigate numerical patterns.

• Finite Sequence: Contains a limited number of terms.
• Infinite Sequence: Continues indefinitely without an end.

Often, sigma notation is used to express sequences succinctly, especially when it’s involved in summation.

In the context of our original exercise, the sequence consists of simply repeating the number 4. This is a very straightforward type of sequence, where every term is identical. However, sequences can be more complex, with terms that vary based on position within the sequence (like arithmetic or geometric sequences). Understanding the pattern of a sequence is key to working effectively with summation.

Mathematical Series

A mathematical series is what you get when you sum the terms of a sequence. Thus, any series can be thought of as the sum of the sequence it originates from.

For example, if you have a finite sequence like {4, 4, 4, 4, 4}, the series derived from this sequence is \(4 + 4 + 4 + 4 + 4\), which equals 20. Mathematical series can come in different types:

• Arithmetic Series: The difference between consecutive terms is constant.
• Geometric Series: Each term is a fixed multiple of the previous term.

In the exercise, the sequence was a constant, so its series was straightforward to calculate. But in cases with arithmetic or geometric series, specialized formulas can simplify the process. Series are not only important in pure mathematics, but also in fields such as finance, physics, and statistics, as they help to understand and predict behavior by analyzing trends in sums of data.