Problem 11 - College Algebra with Corequisite Support

Question

For the following exercises, express each arithmetic sum using summation notation. \(10+18+26+\ldots+162\)

Step-by-Step Solution

Answer

\[\sum_{i=1}^{20} (10 + (i-1)\cdot 8)\]

Step 1: Identify the Sequence

Observe the pattern in the series: 10, 18, 26, ..., 162. Notice that each term increases by a constant amount. This indicates that the series is an arithmetic sequence.

Step 2: Find the Common Difference

To find the common difference \(d\), subtract the first term from the second term: \(18 - 10 = 8\). Thus, \(d = 8\).

Step 3: Determine the Number of Terms

Use the formula for the \(n\)-th term of an arithmetic sequence \(a_n = a_1 + (n-1)\cdot d\). Here, the last term \(a_n = 162\) and the first term \(a_1 = 10\). Substitute the known values and solve for \(n\):\[162 = 10 + (n-1)\cdot 8\]\[152 = (n-1)\cdot 8\]\[n-1 = \frac{152}{8} = 19\]\[n = 20\] Hence, there are 20 terms in the series.

Step 4: Write the Summation Notation

Summation notation expresses the full series compactly. The general term \(a_i\) can be expressed as \(a_i = 10 + (i-1)\cdot 8\). We can write the sum as:\[\sum_{i=1}^{20} (10 + (i-1)\cdot 8)\]

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms remains constant. This type of sequence is characterized by its simplicity and consistent pattern.
Recognizing an arithmetic sequence helps simplify problems by allowing the use of simple formulas for different calculations. In the sequence provided, namely, 10, 18, 26, ..., 162, notice how each number increases by the same value from the previous one, which indicates that it is an arithmetic sequence.

• An arithmetic sequence often takes the form: either \(a_1, a_2, a_3, \ldots\) \ or \(a_1, a_1 + d, a_1 + 2d, \ldots\).

The constant difference forms the crux of the sequence and determines the subsequent term when one term is known.

Common Difference

The common difference in an arithmetic sequence is the consistent amount that each term increases by. Calculating the common difference correctly is crucial as it affects the entire pattern of the sequence.
In this exercise, you found the common difference by subtracting the first term from the second term.

• The formula for a common difference, \(d\), is given by: \(d = a_2 - a_1 \ \)
• For the sequence, \(10, 18, 26, \ldots\), calculate \(d\) as follows: \(18 - 10 = 8\).

An understanding of the common difference allows you to determine subsequent terms and solve related problems accurately.

Number of Terms

Determining the number of terms in an arithmetic sequence involves finding how far the sequence extends. Knowing this helps you define the limits for summation or any operation involving the sequence.
To find the number of terms, you can use the formula for the \(n\)-th term:

• Given by \(a_n = a_1 + (n-1) \cdot d\), where:
• \(a_n\) is the last term,
• \(a_1\) is the first term, and
• \(n\) is the number of terms.

Substitute the known values: \(162 = 10 + (n-1) \cdot 8\), to find that \(n = 20\).
Thus, there are 20 terms in the arithmetic sequence. Knowing \(n\) is essential to understanding the full series.

General Term Formula

The general term formula of an arithmetic sequence allows any term in the sequence to be found without listing all preceding terms. This formula can save time and effort and is pivotal for solving and expressing series.
The general term formula for an arithmetic sequence is expressed as:

• \(a_i = a_1 + (i-1)\cdot d\), where:
• \(a_i\) represents any term in the sequence,
• \(a_1\) is the first term of the sequence, and
• \(i\) is the position of the term in the sequence.

For our sequence, the general term can be expressed as: \(a_i = 10 + (i-1)\cdot 8\).
This expression formulates a powerful way to describe and compute the sequence quickly. Especially when dealing with summation, this formula helps write sequence information compactly using symbols like \(\Sigma\).