Problem 85 - Precalculus with Limits

Question

In Exercises 85-96, find the sum. \( \displaystyle \sum_{i=1}^{5} (2i + 1) \)

Step-by-Step Solution

Answer

The sum of the given sequence from 1 to 5 is 35.

Step 1: Understanding the Summation

The exercise uses the Greek capital letter sigma \(\Sigma\) for defining a sum. The sum \( \displaystyle \sum_{i=1}^{5} (2i + 1) \) means that we are summing the terms of the sequence \(2i + 1\) for \(i\) ranging from 1 to 5.

Step 2: Substitution

Substitute every integer from 1 to 5 in the sequence and sum them all together. This gives \(2*1 + 1 + 2*2 + 1 + 2*3 + 1 + 2*4 + 1 + 2*5 + 1\).

Step 3: Simplification

This simplifies to \(3 + 5 + 7 + 9 + 11\). Summing these terms together yields \(35\).

Key Concepts

Series

A series refers to the sum of the terms in a sequence. In mathematics, when we "sum," we add the terms together. It's a fundamental concept that helps connect sequences to broader mathematical themes. A series can be either finite or infinite.

• In a finite series, there are a specific number of terms to be added. In our case, the series happens over the interval from 1 to 5.
• An infinite series continues indefinitely, often requiring special methods to find its sum.

In our original exercise, we dealt with a finite series since we summed up terms from 1 to 5 of the expression \(2i + 1\). Each expression value was a term in this series.
When working with series, always start by identifying the pattern or rule forming the sequence of numbers for the terms. This pattern helps you know what to add together. In the exercise, the rule was specified as \(2i + 1\). Observe how easy it is to substitute and calculate each term using this rule.
Mastering the concept of series not only helps with sequences but is key in calculus and other advanced mathematics. This fundamental concept builds a foundation for broader mathematical comprehension.

Arithmetic Sequence

An arithmetic sequence is a list of numbers where each term after the first is the result of adding a constant difference to the previous term. It is one of the most commonly encountered sequences in mathematics.

• The common difference might be positive, negative or zero.
• In our exercise, the sequence of numbers generated from \(2i + 1\), was \(3, 5, 7, 9, 11.\)

In this sequence, you can easily spot that the common difference is 2. By observing the way each term increases from the last, you identify this pattern: 5 comes after 3 by adding 2, 7 comes after 5, and so on.
Arithmetic sequences are significant because they provide neat, predictable sets of numbers. They make calculations involving large sums more manageable. Understanding how an arithmetic sequence functions allows you to predict its terms without manually calculating each one. Formulas can derive specific terms or even entire sums. In this case, using the formula for the \(n\)th term of an arithmetic sequence to identify terms quickly would be helpful. Applied correctly, these principles ease your ability to sum large lists of numbers efficiently.

Sigma Notation

Sigma notation is a compact way to express the sum of a sequence defined by a rule or formula. In the context of mathematics, \(\Sigma\) is the capital letter for 'S', indicating 'Sum'. This notation is powerful in simplifying expressions.

• The expression \( \sum_{i=1}^{5} (2i + 1) \) illustrates Sigma notation.
• It concisely indicates the numbers from 1 to 5 should be used as values for \(i\), input into the expression \(2i + 1\).

You substitute each integer in this sequence into the rule \(2i + 1\). This process yields each term of the series. It's an efficient method especially suited for larger sets of numbers or more complicated expressions.
With Sigma notation, appearing as \( \sum \), you can express complex sequences simply and capture essence in a compact form. This becomes particularly useful in calculus, sequences, and series. As with our exercise, Sigma notation elegantly represents the operation of a long sequence, making deriving specific results like sums easier. Mastering it is essential for anyone delving deeper into mathematics as it’s frequently used across various fields.