Problem 13 - Calculus

Question

Write the indicated sum in sigma notation. \(a_{1}+a_{3}+a_{5}+a_{7}+\cdots+a_{99}\)

Step-by-Step Solution

Answer

\( \sum_{n=1}^{50} a_{(2n-1)} \)

Step 1: Understanding Sigma Notation

Sigma notation is a way to write sums that have a regular pattern or sequence. In this case, the terms appear to be part of an arithmetic sequence where only the odd numbered terms are summed.

Step 2: Identifying the Pattern

The terms in the sequence are given by the index of odd numbers. We have the sequence indices as 1, 3, 5, 7, ..., 99. Notice that these are all the odd numbers from 1 to 99, which can be expressed using an arithmetic sequence.

Step 3: Finding the General Form

The general form of the index of the sequence is given by the formula for generating odd numbers: the -th odd number can be expressed as 2n - 1, where n starts from 1.

Step 4: Express the Series in Sigma Notation

Using sigma notation, the series can be expressed as:\[\sum_{n=1}^{50} a_{(2n-1)}\] This is because when n=1, the term is \(a_1\), and when n=50, the term is \(a_{99}\).

Key Concepts

Sigma Notation

Sigma notation is a powerful mathematical tool used to concisely write sums of sequences. The notation employs the Greek letter sigma (\(\sum\)) to indicate summation. This enables mathematicians and students to quickly and clearly represent the addition of elements in a series without having to write each term individually. In sigma notation, the expression to sum is written to the right of the sigma symbol, while the bounds of the sum are placed below and above the symbol. For example, in the notation \[ \sum_{n=1}^{k} b_n \],

• \(n\) is the index of summation, which starts from 1 and increases by 1 for each term.
• \(k\) is the upper limit, indicating the last value of \(n\) for which the terms are added.
• \(b_n\) represents the general term of the sequence being summed.

Sigma notation is especially useful in calculus and higher mathematics due to its ability to succinctly represent extensive calculations, making it a favored technique in mathematical expressions.

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which each term after the first is generated by adding a constant difference to the previous term. This constant is known as the common difference. Arithmetic sequences are linear and predictable, making them fundamental in mathematics.

Key characteristics of an arithmetic sequence include:

• Each term can be expressed in relation to the previous term plus a fixed value: \(a_{n} = a_{1} + (n-1)d\), where \(d\) is the common difference.
• The sequence is linear, which means it progresses in a straight line when plotted on a graph.
• Arithmetic sequences are used in various fields, from calculating interest to analyzing patterns in data.

In the context of the original exercise, only odd-numbered indices were part of the sequence, resulting in the formula for odd numbers: \(2n - 1\). This illustrates how arithmetic sequences can model specific patterns by careful selection of their parameters and initial terms.

General Form

The general form of a sequence or series provides a formula to find any term within the sequence without listing all the previous terms. This form is essential for understanding and working with sequences in mathematics.

When dealing with arithmetic sequences, the general form helps identify any specific term through a straightforward formula. For an arithmetic sequence:\[ a_{n} = a_{1} + (n-1)\cdot d \]

• \(a_{n}\) is the \(n\)-th term you're seeking.
• \(a_{1}\) is the first term in the sequence.
• \(d\) is the common difference between terms.

The general form is particularly useful in expressing sequences in sigma notation, as it precisely defines the transformation from one term to the next. It enables calculating sums through their sequence indices, as seen in the step-by-step solution using the indices \(2n-1\) to target only odd terms from 1 to 99.