Problem 1 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the value of the indicated sum. $$ \sum_{k=1}^{6}(k-1) $$

Step-by-Step Solution

Answer

The value of the sum is 15.

Step 1: Understand the Summation

The problem asks us to find the sum \[\sum_{k=1}^{6}(k-1)\] which means we need to calculate the value by substituting values from \( k = 1 \) to \( k = 6 \) into the expression \( k-1 \) and then add the results together.

Step 2: Generate Sequence Terms

Substitute each value of \( k \) from 1 to 6 into the expression \( k-1 \):- For \( k = 1 \), we get \( 1 - 1 = 0 \).- For \( k = 2 \), we get \( 2 - 1 = 1 \).- For \( k = 3 \), we get \( 3 - 1 = 2 \).- For \( k = 4 \), we get \( 4 - 1 = 3 \).- For \( k = 5 \), we get \( 5 - 1 = 4 \).- For \( k = 6 \), we get \( 6 - 1 = 5 \).

Step 3: Summation of Sequence Terms

Add the terms generated in Step 2: \[0 + 1 + 2 + 3 + 4 + 5\].

Step 4: Calculate the Total Sum

Add the sequence terms together: \[0 + 1 = 1,\]\[1 + 2 = 3,\]\[3 + 3 = 6,\]\[6 + 4 = 10,\]\[10 + 5 = 15.\]The sum \( \sum_{k=1}^{6}(k-1) \) is \( 15 \).

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the "common difference." For example, in the exercise given, when you plug in consecutive values of \( k \) into \( k-1 \), you get the series of numbers \( 0, 1, 2, 3, 4, 5 \).

• The first term here is 0.
• Each successive term increases by 1, which is the common difference.

Understanding arithmetic sequences is essential in this exercise because it helps you recognize patterns and understand how the sequence of numbers will evolve. In other scenarios, arithmetic sequences can help solve everyday problems, like calculating payment schedules or predicting certain patterns.

Series

A series is the sum of the terms of a sequence. When we talk about a series, we're interested in adding up all the terms in the sequence.

In the provided exercise, after generating the sequence \( 0, 1, 2, 3, 4, 5 \), we need to add these terms up to find the series' sum. Performing this calculation, you get:

• \( 0 + 1 + 2 + 3 + 4 + 5 = 15 \)

This result is the sum of an arithmetic sequence. Recognizing this can provide an efficient way to check our work or apply a formula directly in more complex sequences.

Summation Notation

Summation notation, often represented by the symbol \( \sum \), provides a compact way to express long sums. Instead of writing out each term in addition, you can express the operation succinctly.

In the exercise, the notation \( \sum_{k=1}^{6}(k-1) \) tells us to evaluate the expression \( k-1 \) from \( k=1 \) to \( k=6 \) and sum it up.

The general format of summation notation is:

• \( \sum_{k=a}^{b} f(k) \)
• where \( k \) is the index of summation, \( a \) is the lower bound, \( b \) is the upper bound, and \( f(k) \) is the expression to evaluate.

This powerful notation helps not only in simplifying expression elaboration on paper, but also when coding or discussing problems efficiently with others.

Discrete Mathematics

Discrete Mathematics is a branch of mathematics focused on distinct and separate values, often involving integers or finite sets. Unlike calculus which deals with continuous data, discrete mathematics is concerned with individual objects and their interactions.

Summation and sequences are crucial concepts within this discipline, as they deal with countable and whole numbers. In our specific exercise, we deal with a discrete set of numbers \( 1, 2, 3, 4, 5, 6 \) where each value is distinct and separate.

Applications of discrete mathematics span computer science, information theory, and cryptography. Understanding discrete concepts like sequences and series can thus be exceptionally valuable in these fields.