Question

In Exercises \(13-16,\) write the sum using sigma notation. $$ 4+8+12+16+20+24 $$

Short answer

Express the sum 4 + 8 + 12 + 16 + 20 + 24 using sigma notation. Solution: The sum can be represented using sigma notation as: $$ \sum_{n=1}^{6} (4 + (n - 1)4) = \sum_{n=1}^{6} (4n) $$

Step-by-step solution

Identify the pattern in the sum

In the given sum, the terms are: $$ 4, 8, 12, 16, 20, 24 $$ We can see that each term is obtained by adding \(4\) to the previous term, which shows that we have an arithmetic sequence.

Find a general formula for the arithmetic sequence

Write down a general formula for this arithmetic sequence. Let the nth term be denoted as \(a_n\). Since the terms are increasing by 4 from the beginning, we get: $$ a_n = a_1 + (n - 1)d $$ where \(a_1 = 4\) is the first term and \(d = 4\) is the common difference. Substituting these values, the general formula becomes: $$ a_n = 4 + (n - 1)4 $$

Write the sum using sigma notation

Now, we can express the given sum using the sigma notation. We know that the sum contains 6 terms, so n varies from 1 to 6. The sigma notation is given by: $$ \sum_{n=1}^{6} a_n $$ Substituting the general formula for \(a_n\) from Step 2, we get: $$ \sum_{n=1}^{6} (4 + (n - 1)4) $$ So, the sum can be represented using sigma notation as: $$ \sum_{n=1}^{6} (4 + (n - 1)4) = \sum_{n=1}^{6} (4n) $$

Key concepts

Arithmetic Sequence

An arithmetic sequence is a series of numbers in which each term after the first is formed by adding a fixed, constant number, known as the common difference, to the previous term.
For example, in the series \(4, 8, 12, 16, 20, 24\), the common difference is 4. This means we add 4 to each term to get the next one.
Understanding the pattern or rule that defines the sequence is crucial to solving problems related to arithmetic sequences.
The general formula to find the \(n\)th term \(a_n\) of an arithmetic sequence, given the first term \(a_1\) and the common difference \(d\), is:

• \(a_n = a_1 + (n - 1) \, d\)

Using the formula, we can easily calculate any term in the sequence without listing all previous terms. By applying this formula and identifying the sequence's pattern, we can transition into using sigma (summation) notation for further calculations in sequences.

Sum of Series

The sum of a series refers to the total when all terms in the sequence are added together.
For arithmetic sequences, there's a shortcut to calculate this sum without adding each term individually, which is especially useful for long sequences.
The formula for the sum \(S_n\) of the first \(n\) terms of an arithmetic series is:

• \(S_n = \frac{n}{2} (a_1 + a_n)\)

Where \(a_1\) is the first term, and \(a_n\) is the last term in the series.
This formula is derived from taking the average of the first and last terms and multiplying by the number of terms.
However, you can also use sigma notation as a compact form to represent long summations, which leads us to the next concept.

Sigma Notation Formula

Sigma notation is a mathematical symbol used to express the sum of a series in an abbreviated form.
It is represented by the Greek letter \(\Sigma\), and it's a convenient way to denote the sum of a sequence.
In sigma notation, the formula looks like this:

• \(\sum_{n=1}^{k} f(n)\)

Where:

• \(n\) is the index of summation, representing each term's position in the sequence.
• \(k\) is the upper bound, indicating the number of terms to sum.
• \(f(n)\) is the function defining the term's value at each position.

For example, in the exercise provided, we have the sequence \(4n\), and we want the sum from \(n=1\) to \(n=6\), so it looks like \(\sum_{n=1}^{6} 4n\).
Using sigma notation allows us to handle series sums efficiently, especially as the series grows larger.