Problem 9 - Precalculus

Question

In Exercises \(9-16\), rewrite the sum using summation notation. $$ 8+11+14+17+20 $$

Step-by-Step Solution

Answer

The sum can be rewritten as \(\sum_{n=0}^{4} (8 + 3n)\).

Step 1: Identify the Pattern

Observe the given sequence: \(8, 11, 14, 17, 20\). Notice that each term increases by 3. The sequence can be expressed in terms of a starting value plus multiples of the increment: \(8 + 3n\). Here, \(n\) represents the position of the term starting from 0.

Step 2: Determine the Sequence Formula

Since each term can be expressed as \(8 + 3n\), we need to find a general term formula. Start with \(n = 0\) for the first term, which gives \(8 + 3(0) = 8\). Therefore, the formula for the \(n\)-th term is \(8 + 3n\).

Step 3: Establish the Range of Summation

Identify how many terms are included in the sum. By inspecting, we see the sequence ends at 20. Solving for \(n\) in the equation \(8 + 3n = 20\) gives \(3n = 12\) and thus \(n = 4\). Therefore, \(n\) ranges from 0 to 4.

Step 4: Write in Summation Notation

Using the formula from Step 2 and the range from Step 3, express the sum in summation notation: \(\sum_{n=0}^{4} (8 + 3n)\). This notation represents the sum of the sequence from \(n = 0\) to \(n = 4\), effectively rewriting the entire sum.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers where each term after the first is found by adding a constant difference to the previous term. This constant is known as the common difference. In the given exercise, the sequence is:

• 8, 11, 14, 17, 20

Here, each term increases by 3, making 3 the common difference of this arithmetic sequence.
Identifying the pattern in an arithmetic sequence requires observing the increase or decrease between consecutive terms. Once you detect a consistent pattern, you can express this sequence in terms of its first term and the common difference.
For instance, for the sequence above, we can start with the first term (8) and continuously add the common difference (3) to get subsequent terms. This insight helps in constructing the sequence formula.

Sequence Formula

In an arithmetic sequence, each term can be derived using a specific formula called the sequence formula. This formula is straightforward:

• First term: 8
• Common difference: 3

To find the n-th term of the sequence, you use the formula:

• \[a_n = a_1 + (n-1)d\]

Where:

• \(a_n\) is the n-th term
• \(a_1\) is the first term (8)
• d is the common difference (3)

For this problem, however, we use a slightly modified formula because n starts at 0:

• \[a_n = 8 + 3n\]

This sequence formula allows you to calculate any term in the sequence simply by substituting for n with the term position number. For example, when n=0, the term is 8; for n=1, the term is 11, and so on.

Summation Range

The summation range defines the first and last terms included in the sum when using summation notation. It's crucial to determine this range so that all terms are correctly accounted for in a series.
In this exercise, the pattern or sequence starts with n = 0 when the term is 8, and it ends with n = 4 when the term is 20.
To determine that the last term, n = 4, corresponds to 20, we solve the equation given by the sequence formula:

• \[8 + 3n = 20\]
• Solve for n: \[3n = 12\]
• Therefore, n = 4.

This means the summation range is from n = 0 to n = 4, encompassing a total of 5 terms. Having a clear understanding of the summation range ensures each part of the sequence is accurately summed.

Mathematical Notation

Mathematical notation is the language through which mathematics expresses numbers, operations, and relationships concisely. Summation notation, a part of this language, simplifies the expression of sums.Using summation notation, we rewrite the sequence:

• \[\sum_{n=0}^{4} (8 + 3n)\]

This expression succinctly encapsulates the sum of all terms in the sequence from n = 0 to n = 4. The Greek letter Sigma (\(\sum\)) used here represents the instruction to sum up the expression that follows.Within the notation:

• The lower index (n = 0) shows the start of the sequence.
• The upper index (n = 4) indicates the end of the sequence.
• The expression \(8 + 3n\) inside the summation sign delineates what should be repeatedly added as n iterates through the specified range.

Understanding this notation is like having a powerful, friendly tool in your mathematical toolbox, allowing for more efficient communication and problem-solving in math.