Problem 53 - Precalculus: Mathematics for Calculus

Question

Find the partial \(\operatorname{sum} S_{n}\) of the arithmetic sequence that satisfies the given conditions. $$a=-40, \quad d=14, \quad n=15$$

Step-by-Step Solution

Answer

The partial sum of the sequence is 870.

Step 1: Understanding the Formula

The formula to find the sum of the first \( n \) terms of an arithmetic sequence is: \[S_n = \frac{n}{2} (2a + (n-1)d)\], where \( a \) is the first term, \( d \) is the common difference, and \( n \) is the number of terms. Here, \( a = -40 \), \( d = 14 \), and \( n = 15 \).

Step 2: Substitute Known Values

Let's substitute the given values into the formula: \[S_{15} = \frac{15}{2} (2(-40) + (15-1)\cdot14)\]

Step 3: Calculate Internal Expressions

First, calculate the expressions inside the parentheses: \( 2(-40) = -80 \) and \( (15-1)\cdot14 = 14 \cdot 14 = 196 \).

Step 4: Evaluate the Parentheses

Next, evaluate the sum inside the parentheses: \(-80 + 196 = 116\).

Step 5: Complete the Sum Formula Calculation

Substitute back the value into the sum formula: \[S_{15} = \frac{15}{2} \cdot 116 = \frac{15 \cdot 116}{2}\]Now multiply: \(15 \times 116 = 1740\), and divide by 2 to get:\( \frac{1740}{2} = 870 \).

Key Concepts

Partial Sum

The concept of a partial sum in an arithmetic sequence refers to the sum of a certain number of terms from the sequence, rather than the sum of all the terms. Understanding how to calculate a partial sum is useful when you want to find the cumulative total for only a section of the entire sequence. In arithmetic sequences, the partial sum of the first \( n \) terms is calculated using a specific formula which incorporates the first term and the common difference of the sequence.

• The partial sum formula for an arithmetic sequence is: \[ S_n = \frac{n}{2} (2a + (n-1)d) \]
• \( S_n \) stands for the sum of the first \( n \) terms.
• This formula is derived from the fact that when you add terms from the beginning and the end of an arithmetic sequence, the sums are constant.

Applying this formula correctly allows us to find the partial sum for any given number of terms in the series.

Common Difference

In an arithmetic sequence, the common difference is a crucial component. It refers to the consistent difference between consecutive terms in the series.This difference indicates by how much each term increases or decreases as you progress through the sequence.

• Denoted as \( d \) in formulas
• The common difference is calculated by subtracting any term from the subsequent term in the sequence.
• If \( d \) is positive, the sequence is increasing; if \( d \) is negative, it is decreasing.

Understanding the common difference helps determine the pattern and progression of the sequence, and is essential for calculating the sum or any particular term effectively.

First Term

The first term of an arithmetic sequence is simply the very first number in the sequence, denoted by \( a \).It is one of the initial values you need to identify in order to use most arithmetic sequence formulas.

• \( a \) is always the starting point of the sequence
• It sets the baseline from which the sequence progresses based on the common difference.

For instance, in our sequence where \( a = -40 \), it tells us that initially, the sequence begins at \(-40\).Identifying \( a \) allows you to plug it into various formulas to find other characteristics of the sequence.

Number of Terms

The number of terms in an arithmetic sequence, denoted as \( n \), indicates how many terms you are considering or summing up from the sequence.It is a simple count that plays a critical role in calculating the partial sum.

• Serves as a boundary for calculations in sequence formulas
• \( n \) determines how many terms you need to account for when using the partial sum formula \( S_n \)
• In our example, \( n = 15 \), which means we are interested in the sum of the first 15 terms.

Without knowing the number of terms, it would be impossible to calculate the partial sum, making \( n \) essential for holistic sequence analysis.