Problem 13 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Pattern

First, notice that the indices in the sequence are odd numbers: 1, 3, 5, 7, ..., 99. This suggests that the indices are increasing by 2 each time.

Step 2: Determine the General Term

The general term for this series can be described by the formula for odd numbers: \(a_{2k-1}\), where \(k\) represents the term number in the sequence of odd numbers.

Step 3: Identify the First and Last Terms

The first term when \(k = 1\) is \(a_1\), and the last term in the series is \(a_{99}\) which corresponds to odd number 99. To find the index \(k\) for this last term, solve \(2k-1 = 99\).

Step 4: Solve for k in Last Term

Solving \(2k-1 = 99\) gives \(2k = 100\), so \(k = 50\). This means there are 50 terms in the sequence.

Step 5: Write Sigma Notation

Write the sum using sigma notation: \(\sum_{k=1}^{50} a_{2k-1}\). This notation compactly represents the sum of terms with odd indices from the first term up to the 50th term.

Key Concepts

Series Sequence

A series sequence is a list of numbers where each term can be defined based on the position in that list. In the given exercise, we have a sequence deriving from a series of odd-numbered indices: \(a_{1}, a_{3}, a_{5}, \ldots, a_{99}\). This highlights that a pattern is present—in this case, the indices increase by 2, making them all odd. Such sequences are foundational in understanding how series can be broken down into manageable parts with clear, repeating structures.

• Every number in the sequence has a position and a value. This position-based approach is key to identifying sequences.
• Examples of sequences include even numbers, squares, cubes, or in this case, sequences with specific odd indices.

Recognizing and understanding the series sequence is crucial. It allows us to predict and comprehend the behavior of any series based on established rules or patterns.

Sum of Odd Indices

When calculating the sum of a subset of a series—such as terms positioned at odd indices—it's essential to establish the rule governing these positions. In the exercise, the series involves only those terms where the indices are odd numbers. Rather than dealing with the entire series, we focus on:

• The pattern among odd indices (i.e., 1, 3, 5, ..., 99).
• How these indices contribute to determining the sum.

It simplifies calculations by reducing the terms we need to sum up while maintaining a consistent rule for identifying valid terms. By calculating only these specific positions, it reflects the inner symmetry and properties of series operations.

General Term Formula

Finding a general term formula helps in expressing any specific term within a sequence without having to enumerate each term individually. For the odd indices in this task, we use the formula \(a_{2k-1}\) where \(k\) denotes the position in the subset of the sequence. This is derived from the observation:

• Odd indices progress as \(1, 3, 5, \ldots, 99\), showing an arithmetic progression.
• The formula \(2k-1\) captures this progression, ensuring each term is odd.

Calculating exactly where a particular term lies within the sequence becomes easier. Such formulas are invaluable in simplifying complex summations and understanding broader mathematical constructs.

Mathematical Patterns

Mathematical patterns form the backbone of many combinatorial and numerical operations. Recognizing these patterns allows for the prediction and simplification of calculations. The exercise reveals a valuable insight into identifying the pattern in indexing. Consider these aspects:

• Patterns help in seeing beyond the apparent randomness by providing a solid structure.
• By detecting the odd/even or alternating pattern, tasks become more approachable and solvable.

Understanding such mathematical patterns allows us to write elegant solutions such as the sigma notation used here: \(\sum_{k=1}^{50} a_{2k-1}\). This compact representation captures the essential aspects of the sequence, empowering us to deal effectively with sums, sequences, and more complex mathematical expressions.