Problem 22 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 19-22: Find the sum \(S_{n}\) of the arithmetic sequence that satisfies the stated conditions. $$ a_{7}=\frac{7}{3}, \quad d=-\frac{2}{3}, \quad n=15 $$

Step-by-Step Solution

Answer

The sum of the sequence, \( S_{15} \), is 25.

Step 1: Recall Arithmetic Sequence Formula

The general term of an arithmetic sequence is given by the formula \( a_n = a_1 + (n-1) \cdot d \), where \( a_n \) is the nth term, \( a_1 \) is the first term, \( d \) is the common difference, and \( n \) is the term number.

Step 2: Find the First Term

We know that \( a_7 = \frac{7}{3} \) and \( d = -\frac{2}{3} \). Substitute these into the formula for the 7th term: \( a_7 = a_1 + 6 \cdot d \). Thus, \( \frac{7}{3} = a_1 + 6 (-\frac{2}{3}) \). Solve this to find \( a_1 \): \( a_1 = \frac{7}{3} + 4 = \frac{19}{3} \).

Step 3: Sum of Arithmetic Sequence Formula

The sum of the first \( n \) terms of an arithmetic sequence is given by the formula \( S_n = \frac{n}{2} (a_1 + a_n) \). We need to find \( a_{15} \) to use this formula.

Step 4: Find the 15th Term

Use the formula for the general term to find \( a_{15} \): \( a_{15} = a_1 + (15-1) \cdot d = \frac{19}{3} + 14 \cdot (-\frac{2}{3}) = \frac{19}{3} - \frac{28}{3} = -\frac{9}{3} = -3 \).

Step 5: Calculate the Sum

Plug the values into the sum formula: \( S_{15} = \frac{15}{2} (\frac{19}{3} + (-3)) = \frac{15}{2} \cdot (\frac{19}{3} - \frac{9}{3}) = \frac{15}{2} \cdot \frac{10}{3} = \frac{150}{6} = 25 \).

Key Concepts

Sum of Arithmetic Sequence

To find the sum of an arithmetic sequence, use the formula:

• \( S_n = \frac{n}{2} (a_1 + a_n) \)

This formula tells us how to calculate the total of the first \( n \) terms, where \( S_n \) represents the sum, \( n \) stands for the number of terms, \( a_1 \) is the first term, and \( a_n \) is the last (or nth) term of the sequence.
Begin by finding the first term and the nth term using the sequence's starting conditions. Then plug these values into the formula to calculate the sum. This approach allows you to quickly determine the cumulative effect of the entire sequence within a specified range.
For example, in the problem given, after calculating \( a_1 \) and \( a_{15} \), you can use:

• \( S_{15} = \frac{15}{2} \times (\frac{19}{3} + (-3)) = 25 \)

Indicating the final sum of the sequence is 25.

Common Difference

The common difference \( d \) in an arithmetic sequence is the factor by which the sequence progresses from one term to the next. It's constant and can be calculated by subtracting any term in the sequence from its succeeding term.
Mathematically, if \( a_1 \), \( a_2 \), \( a_3 \), … represent the sequence, then \( d = a_2 - a_1 = a_3 - a_2 \).
In the given exercise, the common difference is provided as \( d = -\frac{2}{3} \). This negative value means that each term in the sequence is decreasing by \( \frac{2}{3} \) compared to the previous term. Understanding this attribute of arithmetic sequences helps in forecasting future terms without computing each one after the other.
Why is this important? It shows how much the sequence changes as it progresses and influences the overall sum when using the sum formula.

General Term Formula

The general term formula helps to find any specific term in an arithmetic sequence without calculating all preceding terms. The formula is given by:

• \( a_n = a_1 + (n-1) \cdot d \)

Here, \( a_n \) is the nth term of the sequence, \( a_1 \) is the first term, \( n \) is the position in the sequence, and \( d \) is the common difference.
By substituting known values into this formula, one can solve for unknown term values as shown in the solution step where \( a_{15} \) was needed.
This formula simplifies the calculation by allowing us to jump directly to any term number without hassle.

First Term of Sequence

Identifying the first term \( a_1 \) of an arithmetic sequence is crucial because it sets the starting point from which all other terms are derived. It is the initial value needed to apply both the general term and the sum of sequences formulas.
In our exercise, the first term was found using information about the 7th term. By utilizing the formula \( a_7 = a_1 + 6 \cdot d \) and given \( a_7 \) and the common difference \( d \), we solved for:

• \( a_1 = \frac{19}{3} \)

This computation shows how earlier terms influence the entire sequence and its sum, making it imperative to understand how to find \( a_1 \) when needed.