Problem 33 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 29-34: Express the sum in terms of summation notation. (Answers are not unique.) $$ \frac{3}{7}+\frac{6}{11}+\frac{9}{15}+\frac{12}{19}+\frac{15}{23}+\frac{18}{27} $$

Step-by-Step Solution

Answer

The sum is \( \sum_{n=1}^{6} \frac{3n}{4n + 3} \).

Step 1: Identify the Pattern in the Numerators

Observe the numerators in the given sum: 3, 6, 9, 12, 15, 18. These numerators form an arithmetic sequence that starts with 3 and increases by 3 with each term. This sequence can be represented by the formula \(3n\), where \(n\) represents the term number starting from 1.

Step 2: Identify the Pattern in the Denominators

Examine the denominators of the fractions: 7, 11, 15, 19, 23, 27. These denominators also form an arithmetic sequence starting from 7 and increasing by 4 for each term. The sequence can be expressed by the formula \((4n + 3)\), where \(n\) starts from 1.

Step 3: Write the General Term of the Sequence

With the explanations from Steps 1 and 2, the general term for the sequence can be given as: \( \frac{3n}{4n + 3} \).

Step 4: Express the Sum Using Summation Notation

The sequence has 6 terms; hence the sum can be expressed using summation notation as: \( \sum_{n=1}^{6} \frac{3n}{4n + 3} \).

Key Concepts

Arithmetic Sequences

An arithmetic sequence is a series of numbers that has a consistent difference between consecutive terms. This difference is called the "common difference." In simple terms, an arithmetic sequence adds the same value to get from one term to the next. For instance, in the sequence 3, 6, 9, 12, 15, 18, each term increases by 3. This makes the common difference 3.

• First term: The starting number, which is 3 in our sequence.
• Common difference: The number added each time, which is also 3 here.

Understanding arithmetic sequences helps in predicting additional terms, identifying patterns, and preparing expressions for calculations.

General Term of a Sequence

The general term of a sequence is the expression that allows you to find any term within the sequence without listing them all. For an arithmetic sequence, the general term can be derived using the formula:\[ a_n = a_1 + (n - 1)d \]Where:

• \(a_n\): The n-th term you want to find.
• \(a_1\): The first term in the sequence, which is 3 in this context.
• \(d\): The common difference, also 3 here.

This formula is essential because it helps identify any term's position in the sequence. To calculate, plug in the first term and the common difference. For example, substituting into the formula, the expression becomes \( a_n = 3n \) for the numerators. Similarly, a similar process helps in identifying the pattern for the denominators, formulated as \((4n + 3)\).

Summation Notation in Algebra

Summation notation is a technique in algebra used to denote the sum of a sequence's terms. It is often represented by the Greek letter Sigma (\( \Sigma \)). This symbol helps write long sums in a compact form.

• The expression under the Sigma tells you which terms to sum up.
• The numbers on the Sigma's top and bottom indicate the range, or start and end term numbers.

For instance, in our problem, the sum of the sequence could be expressed as:\[ \sum_{n=1}^{6} \frac{3n}{4n + 3} \]This notation signifies that we sum the sequence's terms from the first term, \(n=1\), to the sixth term, \(n=6\). This form is efficient and clear, avoiding having to list out all terms separately.