Problem 30 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 29-34: Express the sum in terms of summation notation. (Answers are not unique.) $$ 3+8+13+18+23 $$

Step-by-Step Solution

Answer

\( \sum_{n=1}^{5} (3 + (n-1) \times 5) \)

Step 1: Identify the Pattern

First, observe the sequence of numbers: 3, 8, 13, 18, 23. Notice the pattern or rule. Each number in the sequence is obtained by adding 5 to the previous one.

Step 2: Write the General Formula

We can express the sequence in terms of a general formula. The first term is 3, and each subsequent term is 5 more than the previous one. Thus, the nth term can be represented as: \[ a_n = 3 + (n-1) \times 5 \] where \( n \) represents the position of the term in the sequence.

Step 3: Convert to Summation Notation

The sum of the sequence can be expressed using summation notation. Since there are 5 terms in this sequence, the sum can be written as: \[ \sum_{n=1}^{5} \left(3 + (n-1) \times 5\right) \] This notation indicates that you sum the expression \( 3 + (n-1) \times 5 \) from \( n = 1 \) to \( n = 5 \).

Key Concepts

Understanding Arithmetic Sequences

When you encounter a series of numbers like 3, 8, 13, 18, and 23, what you're looking at is an arithmetic sequence. This type of sequence is characterized by the fact that each successive term is obtained by adding a constant difference to the previous term.

• In this sequence, we start with the first number, 3.
• We notice that each subsequent number is formed by adding 5 to the previous number.

The number 5 here is called the "common difference." It's what differentiates one term from the next in the sequence. Recognizing this pattern is an essential first step in applying summation notation or even understanding sequences in general. By identifying this common difference, we can easily predict or continue the sequence by simply continuing to add 5 to the last known number.

Recognizing Sequence Patterns

Finding the pattern in a sequence is like cracking a code. It allows us to describe the series in mathematical terms, which is particularly useful when dealing with longer sequences.

• To decode the sequence 3, 8, 13, 18, 23, we see that it begins at 3 and increases by 5 each time.
• Each term can be described using a specific position number, known as 'n'.

This sequence pattern helps us derive what's known as the general formula. For an arithmetic sequence like this, the general formula expresses the nth term (the position of interest in the sequence) based on its relationship to the first term and the common difference. As a result, it gives us the power to find any term within the sequence without having to list all preceding numbers.

Deriving the General Formula

The magic of sequences really comes to life when we talk about the general formula. With just a few known terms, this formula allows us to calculate any term in the sequence. For our particular sequence, the general formula is derived as follows:

• The first term is denoted by 3.
• If we call "n" the term's position, and knowing the common difference is 5, we can say:

To derive the nth term,\[ a_n = 3 + (n-1) \times 5 \]Here's how it breaks down:

• 3 is the initial term.
• (n-1) represents the number of "jumps" or steps from the first term.
• Those jumps are each 5 units long because 5 is our common difference.

This formula is immensely powerful because it enables us to extend or explore the sequence far beyond the initial terms we started with. Plus, it lays the groundwork for using summation notation to find the sum of sequences, by summing over this formula from the first term to the last term.