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 sum is 15.

Step 1: Understand the Sum Expression

The given expression is \( \sum_{k=1}^{6}(k-1) \). This summation means we start at \( k = 1 \) and go up to \( k = 6 \), calculating \( k-1 \) for each value of \( k \) and then adding all these results together.

Step 2: Calculate Each Term

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

Step 3: Add the Terms

Add all the calculated terms from Step 2 together: \( 0 + 1 + 2 + 3 + 4 + 5 = 15 \).

Step 4: State the Result

The sum \( \sum_{k=1}^{6}(k-1) \) evaluates to 15.

Key Concepts

Arithmetic Series

An arithmetic series is a sequence of numbers in which the difference between consecutive terms remains consistent. This difference is known as the "common difference." For example, if you have a series like 2, 4, 6, 8, each has a common difference of 2. In arithmetic series, you often sum these terms to find either a specific term or the total of all terms within a certain range.

• To find the sum of an arithmetic series, two key formulas often used are: sum formula and the n^th term formula.
• The sum formula helps you calculate the total of all terms in the series, and it is expressed as: \[ S_n = \frac{n}{2} (a + l) \], where \( \text{S}_n \) is the sum of the first \( n \) terms, \( a \) is the first term, and \( l \) is the last term.
• The n^th term formula is \( a_n = a + (n-1)d \), where \( d \) is the common difference.

When working with arithmetic series, identifying the first term and the common difference quickly can make calculations easier, especially for longer series.

Summation Notation

Summation notation, also known as sigma notation, is a concise way to represent the summation of terms. It uses the Greek letter Sigma (\( \Sigma \)) to represent the sum of a series of terms, expressed as:\[ \sum_{k=1}^{n} a_k \], where \( a_k \) represents the general term for each instance of \( k \).

• The number below the sigma sign, \( k=1 \), tells you where to begin summation.
• The number above the sigma, \( n \), provides the endpoint.
• The expression next to Sigma (e.g., \( k-1 \)) tells you what to evaluate for each increment of \( k \).
• This notation helps simplify complex summations and makes it easy to understand the range and calculation required.

Using summation notation is particularly beneficial in mathematics because it reduces the space needed to write long sums and makes it straightforward to recognize the pattern or rule used in the summation sequence.

Calculating Sums

Calculating the sum of a series involves evaluating each term based on the rule given, then adding these evaluated terms together. This involves these primary steps:
1. **Identify** the Range: Determine the range of terms to sum, in the form of starting and ending values, such as 1 to 6.2. **Evaluate** Each Term: Take each value of the index (e.g., \( k \)) over the range and substitute it into the general term expression (like \( k-1 \)).3. **Add** the Evaluated Terms: Once all terms are evaluated, simply add them up to find the total.

• In the example \( \sum_{k=1}^{6}(k-1) \), you calculate \( k-1 \) for each \( k \) from 1 to 6, resulting in the sequence: 0, 1, 2, 3, 4, 5.
• The sum of these numbers is 15, which represents the total of the sequence evaluations.

Understanding these steps helps effectively tackle problems involving summation notation and arithmetic series, giving a reliable way to find a sum's value in various mathematical contexts.