Problem 16 - Algebra 2 and Trigonometry

Question

In \(15-26,\) write each series in sigma notation. $$ 1+6+11+16+21+26+31+36 $$

Step-by-Step Solution

Answer

The series in sigma notation is \(\sum_{n=1}^{8} (5n - 4)\).

Step 1: Recognize the Pattern

Identify the pattern in the series. Notice that each term increases by 5 from the previous term: \(1, 6, 11, 16, 21, 26, 31, 36\). This is an arithmetic sequence with the first term \(a_1 = 1\) and a common difference \(d = 5\).

Step 2: Determine the General Formula for the nth Term

The general formula for the nth term of an arithmetic sequence is \(a_n = a_1 + (n-1) \cdot d\). Substitute the known values \(a_1 = 1\) and \(d = 5\) to get: \[a_n = 1 + (n-1) \times 5 = 5n - 4\]

Step 3: Count the Terms in the Series

Count the total number of terms. Observing the series \(1, 6, 11, 16, 21, 26, 31, 36\), there are 8 terms.

Step 4: Write the Series Using Sigma Notation

Use the sigma notation to write the series which sums up the terms from \(n = 1\) to \(n = 8\) using the formula for the nth term: \[\sum_{n=1}^{8} (5n - 4)\]

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a list of numbers where each term is obtained by adding a constant difference to the previous term. This constant is known as the "common difference." For example, in the sequence provided in the exercise: 1, 6, 11, 16, 21, 26, 31, 36, every number increases by 5. This consistent increase is what defines the sequence as arithmetic.

• The first term here is 1.
• Each term is 5 units more than the one preceding it.
• This predictable pattern makes arithmetic sequences easy to analyze and work with.

Recognizing such patterns can help in identifying arithmetic sequences just like in our exercise.

General Formula for nth Term

The general formula to find any term in an arithmetic sequence is given by:\[ a_n = a_1 + (n-1)d \]where:

• \(a_n\) represents the nth term you want to find.
• \(a_1\) is the first term of the sequence.
• \(d\) is the common difference between consecutive terms.
• \(n\) is the term number in the sequence.

For the series in our exercise, substituting \(a_1 = 1\) and \(d = 5\) into the formula gives:\[ a_n = 1 + (n-1) imes 5 = 5n - 4 \]This formula allows you to find any term in the sequence, whether it be the 2nd, 10th, or even the 100th term.

Series

A series is the sum of the terms of a sequence. It takes a sequence and adds up the numbers accordingly. In our problem, we're summing up the terms:1, 6, 11, 16, 21, 26, 31, 36.
The series can be written in compact form using sigma notation, which provides a concise way to represent the sum of a sequence. This is particularly handy for long sequences.
In our case, using the formula for the nth term \((5n - 4)\), the series is expressed in sigma notation as:\[ \sum_{n=1}^{8} (5n - 4) \]The symbol \(\sum\) signifies that you sum the expression \(5n - 4\) for \(n\) ranging from 1 to 8.

Common Difference

The common difference in an arithmetic sequence is a critical value. It is the amount that each term increases (or decreases) from the previous one. Simply put, it's what you add to any term to get the next term.
In our example, the common difference \(d\) is 5:

• From 1 to 6 is an increase of 5.
• From 6 to 11 is also an increase of 5.
• This pattern continues for all the terms in the sequence.

Identifying the common difference is key to understanding and working with arithmetic sequences, as it plays a pivotal role in the general formula for the nth term and in transforming the sequence into a series through sigma notation.