Problem 9 - Essential Calculus

Question

In Exercises 7 -12, use sigma notation to write the sum. $$ \left[5\left(\frac{1}{8}\right)+3\right]+\left[5\left(\frac{2}{8}\right)+3\right]+\cdots+\left[5\left(\frac{8}{8}\right)+3\right] $$

Step-by-Step Solution

Answer

The sigma notation for the given series is \(\sum_{i=1}^{8}(5\left(\frac{i}{8}\right)+3)\)

Step 1: Identify pattern

First, identify the pattern. For all terms, the pattern appears to increment by \(\frac{1}{8}\) each time. For instance, the second term for the first summation is \(\frac{1}{8}\), then \(\frac{2}{8}\) for the second, and so forth.

Step 2: Formulate Sigma Notation

The sigma notation is represented as \(\sum_{i=m}^{n} a_i\), where \(i\) is the index of summation, \(m\) and \(n\) are the lower and upper limits of the summation respectively, and \(a_i\) is the ith term. Here, \(i\) moves from 1 to 8, incrementing by \(1/8\) each time. Thus, our series in sigma notation would be \(\sum_{i=1}^{8}(5\left(\frac{i}{8}\right)+3)\)

Key Concepts

Sequence and Series

Understanding sequences and series is crucial for mastering sigma notation and the art of summation. A **sequence** is a list of numbers arranged in a particular order, each number following a specific rule or pattern. These numbers are called terms. In the exercise, we see a set of terms that begin with \( 5\left(\frac{1}{8}\right)+3 \) and continue with each subsequent term adjusting the fraction by a single increment, such as \( 5\left(\frac{2}{8}\right)+3 \).
A **series** is essentially the sum of the terms of a sequence. In other words, when you take multiple terms from a sequence and add them up, you form a series. The exercise asks us to construct a sum of the sequence defined by the series.
Understanding how a sequence becomes a series with the use of sigma notation allows us to manage and simplify arithmetic and geometric progressions effectively.

Summation Index

The summation index in sigma notation acts as a counter that helps us track each term in a series. It is represented as \(i\) in the general formula \(\sum_{i=m}^{n} a_i\).

• The summation index \(i\) starts with the initial value \(m\) and ends at the final value \(n\).


• The index is incremented by 1 with each step unless specified otherwise.

In the given problem, \(i\) starts at 1 and ends at 8. This indicates that there are 8 terms in the sequence that are summed to create the series.
The **summation index** simplifies the representation of long sums by using a concise symbol, making it easier to manage complex arithmetic operations involving numerous terms. Understanding how to set the correct **indexes** is essential for writing and solving series using sigma notation.

Mathematical Patterns

Spotting mathematical patterns is a key skill in writing and understanding series. In the exercise, the pattern can be seen as the fraction \(i/8\) within the expression \5\left(\frac{i}{8}\right)+3\ keeps adding by \1/8\ to the numerator as \(i\) increments.

• This systematic approach where each term increases by a fixed amount, known as the common difference in arithmetic sequences, helps us pinpoint the sequence's rules and apply sigma notation correctly.
• In mathematical notation, recognizing patterns allows us to transform a series of terms into one concise formula.

Using patterns helps in reducing errors and creating simple representations for complex mathematical phenomena. By identifying these structures, you lay a strong foundation for solving not just this problem, but a wide array of mathematical equations and series. Mastering this skill makes dealing with vast sums manageable and often even enjoyable.