Problem 55 - Precalculus: Mathematics for Calculus

Question

Find the partial \(\operatorname{sum} S_{n}\) of the arithmetic sequence that satisfies the given conditions. $$a_{1}=55, \quad d=12, \quad n=10$$

Step-by-Step Solution

Answer

The partial sum \(S_{10}\) of the sequence is 1090.

Step 1: Understand the Problem

The problem asks us to find the partial sum of an arithmetic sequence. We are given the first term \(a_1 = 55\), the common difference \(d = 12\), and the number of terms \(n = 10\).

Step 2: Use the Formula for the nth Term

The formula for the nth term of an arithmetic sequence is \(a_n = a_1 + (n-1) \cdot d\). We will use it to find the 10th term (\(a_{10}\)).

Step 3: Calculate the 10th Term

Plug in the values into the formula: \(a_{10} = 55 + (10-1) \cdot 12 = 55 + 9 \cdot 12 = 55 + 108 = 163\). So, \(a_{10} = 163\).

Step 4: Use the Formula for Partial Sum

The partial sum \(S_n\) of an arithmetic sequence can be calculated using the formula \(S_n = \frac{n}{2} \cdot (a_1 + a_n)\).

Step 5: Plug in the Values to Find the Partial Sum

Using the formula from Step 4, substitute the known values: \(S_{10} = \frac{10}{2} \cdot (55 + 163)\).

Step 6: Calculate the Partial Sum

First, compute inside the parentheses: \(55 + 163 = 218\). Then multiply by \(5\) (which is \(\frac{10}{2}\)): \(S_{10} = 5 \cdot 218 = 1090\).

Key Concepts

Partial Sum

The concept of a partial sum in an arithmetic sequence refers to the sum of a specific number of terms, starting from the first term. In our exercise, we focused on finding the sum of the first 10 terms of the arithmetic sequence. This calculated sum is denoted as \( S_n \), where \( n \) represents the number of terms to be summed. To determine the partial sum, we use the formula for the partial sum of an arithmetic sequence:

• S_n = \frac{n}{2} \cdot (a_1 + a_n)

Here, \( a_1 \) is the first term and \( a_n \) is the nth term. The formula effectively averages the first and last terms and scales it by the number of terms you are adding. This cleverly utilizes the inherent structure of an arithmetic sequence, where each term increases by a common difference. This arithmetic mean approach is efficient and harnesses the power of symmetry in arithmetic progressions.

Common Difference

The common difference is a fundamental component of an arithmetic sequence. In this sequence, each term after the first is obtained by adding a constant value to the preceding term. This constant is known as the common difference, denoted by \( d \). In our example, the common difference is given as \( 12 \).
Understanding and identifying the common difference is crucial because it directly impacts the sequence's behavior. In a sequence:

• If the common difference is positive, as in our case, the sequence is increasing.
• If negative, the sequence decreases.
• If zero, all terms in the sequence are identical.

To calculate any term in the sequence, the formula used is:

• a_n = a_1 + (n-1) \, d

This formula highlights that each term is simply the starting point adjusted by the cumulative addition of the common difference. Knowing the common difference allows us to predictively determine any term in the sequence without sequentially adding the difference each time.

Arithmetic Series

An arithmetic series is fundamentally about summing the terms of an arithmetic sequence. It's an essential concept because it deals with not just listing out terms but evaluating their collective magnitude. This sequence introduces the notion of an arithmetic sum, which uses a set framework to calculate without having to perform laborious addition.
In our scenario, the series begins with the term \( a_1 = 55 \) and follows with additional terms increasing by the common difference of \( 12 \). A series with these terms becomes:

• 55, 67, 79, 91, 103, 115, 127, 139, 151, 163

To find the sum of this series, which is partial due to its limited number of terms, we implement the formula for the partial sum as mentioned previous sections:

• S_n = \frac{n}{2} \cdot (a_1 + a_n)

This formula quickly provides the sum, demonstrating the arithmetic sequence's predictability and simplicity when organizing regular additions of a fixed interval, greatly facilitating comprehension and calculation.