Problem 47 - Algebra and Trigonometry

Question

\(43-48\) . Find the partial sum \(S_{n}\) of the arithmetic sequence that satisfies the given conditions. $$ a_{1}=55, d=12, n=10 $$

Step-by-Step Solution

Answer

The partial sum of the sequence is 1090.

Step 1: Understanding the Problem

The problem asks for the partial sum, denoted as \(S_n\), of an arithmetic sequence. Given are the first term \(a_1 = 55\), the common difference \(d = 12\), and the number of terms \(n = 10\).

Step 2: Formula for Partial Sum of an Arithmetic Sequence

The formula for the partial sum \(S_n\) of the first \(n\) terms of an arithmetic sequence is given by: \[ S_n = \frac{n}{2} (a_1 + a_n) \] where \(a_1\) is the first term and \(a_n\) is the last term, which can be found using \(a_n = a_1 + (n-1) \cdot d\).

Step 3: Calculate the Last Term of the Sequence

Using the formula for the nth term of an arithmetic sequence: \[ a_n = a_1 + (n-1) \cdot d \] Substitute the known values: \( a_1 = 55 \), \( d = 12 \), \( n = 10 \). Therefore, \[ a_{10} = 55 + (10-1) \times 12 = 55 + 108 = 163 \].

Step 4: Calculate the Partial Sum

Now that we have both \(a_1 = 55\) and \(a_{10} = 163\), use the partial sum formula: \[ S_{10} = \frac{10}{2} (55 + 163) \]. Simplify to find \[ S_{10} = 5 \times 218 = 1090 \].

Step 5: Conclusion: Write the Solution

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

Key Concepts

Partial Sum

The partial sum in an arithmetic sequence refers to the sum of the first few terms, specifically the first "n" terms. It's a useful way to quickly calculate the cumulative total for a portion of the sequence.
To find the partial sum, we use the formula:

• \[ S_n = \frac{n}{2} (a_1 + a_n) \]

This formula essentially averages the first and the last term of the sequence and multiplies it by the number of terms. It's incredibly efficient because it avoids the need to add each individual term manually. Given the first term \( a_1 = 55 \), last term \( a_{10} = 163 \), and number of terms \( n = 10 \), we calculated that \( S_{10} = 1090 \).
This provides the sum of the sequence from the first to the tenth term.

Common Difference

The common difference in an arithmetic sequence is the difference between consecutive terms. It's typically denoted by the letter \( d \).
This consistent difference is what makes the sequence "arithmetic". To determine the common difference, you subtract any term in the sequence from the subsequent term. For example, if you have a sequence starting with 55 and each term increases by 12, then the common difference \( d \) is 12.
In our exercise, \( d = 12 \) is crucial for finding any term in the sequence using the nth term formula. This regularity allows you to calculate the value of terms further along the sequence without listing all previous terms.

Nth Term Formula

To find a specific term in an arithmetic sequence, you use the nth term formula. This formula is:

• \[ a_n = a_1 + (n-1) \cdot d \]

Here, \( a_n \) represents the term you want to find, \( a_1 \) is the first term, \( n \) is the position of the term in the sequence, and \( d \) is the common difference. This formula provides a direct way to find any term without listing all prior terms.
In the original exercise, we found the 10th term as follows: with \( a_1 = 55 \), \( n = 10 \), and \( d = 12 \), substitute these values into the formula to get \( a_{10} = 163 \). This calculation is a key step prior to determining the partial sum.

First Term

The first term of an arithmetic sequence, denoted as \( a_1 \), serves as the starting point for the sequence. It's crucial because all other terms are generated based on this starting value and the common difference.
Understanding the role of the first term is fundamental when working with sequences. For this exercise, \( a_1 = 55 \) is the initial value. It sets the stage for how all subsequent terms are determined through the nth term formula and contributes to calculating partial sums.
Without the first term, the rest of the arithmetic sequence cannot be accurately generated. It's like the foundation of a building; everything builds up from it, including how you determine sums or specific terms in the sequence.