Problem 47 - Precalculus Mathematics for Calculus

Question

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 first 10 terms of the sequence is 1090.

Step 1: Understand the Formula for Partial Sum

The partial sum of an arithmetic sequence can be found using the formula \( S_n = \frac{n}{2} \times (2a_1 + (n-1) \times d) \), where \( S_n \) is the partial sum, \( n \) is the number of terms, \( a_1 \) is the first term, and \( d \) is the common difference.

Step 2: Substitute the Given Values

Substitute \( a_1 = 55 \), \( d = 12 \), and \( n = 10 \) into the formula. This gives: \[ S_{10} = \frac{10}{2} \times (2 \times 55 + (10-1) \times 12). \]

Step 3: Simplify Inside the Parentheses

First, compute \( 2 \times 55 = 110 \). Next, calculate \( (10-1) \times 12 = 9 \times 12 = 108 \). Thus,\[ S_{10} = \frac{10}{2} \times (110 + 108). \]

Step 4: Calculate the Sum in Parentheses

Add the numbers inside the parentheses: \( 110 + 108 = 218 \). This gives us:\[ S_{10} = \frac{10}{2} \times 218. \]

Step 5: Simplify the Entire Expression

Divide \( 10 \) by \( 2 \) to get \( 5 \), and then multiply by \( 218 \):\( 5 \times 218 = 1090 \). So, the partial sum \( S_{10} = 1090 \).

Key Concepts

Partial Sum

In the realm of arithmetic sequences, a partial sum, denoted as \( S_n \), is the sum of the first \( n \) terms of the sequence. It represents a portion of the entire sequence, which consists of an infinite number of terms. To put it simply, the partial sum can be imagined as taking a chunk out of this infinite sequence and summing it up.
The formula for the partial sum of an arithmetic sequence is given by:

• \( S_n = \frac{n}{2} \times (2a_1 + (n-1) \times d) \)

Here:

• \( S_n \) is the partial sum
• \( n \) is the number of terms to be summed
• \( a_1 \) is the first term of the sequence
• \( d \) is the common difference between consecutive terms

By plugging in the given values into this formula, one can easily find the desired partial sum. This technique is a straightforward way to manage and compute sums for any sequence efficiently.

Sequence Formula

The sequence formula is like a roadmap for the entire arithmetic sequence, helping you determine any term in the sequence. By knowing this formula, you can find out not just the first few terms, but any term numbered \( n \), even if \( n \) is quite large.
For an arithmetic sequence, the typical formula to find the \( n^{th} \) term \( a_n \) is:

• \( a_n = a_1 + (n-1) \times d \)

Breaking it down:

• \( a_n \) is the \( n^{th} \) term
• \( a_1 \) is the first term, which often serves as the starting point
• \( n \) is the position of the term within the sequence
• \( d \) is the common difference, telling you how much you need to add or subtract to get from one term to the next

Initially, using this formula can seem a bit abstract, but with practice, it becomes a powerful tool for calculating any term in an arithmetic sequence on the fly.

Common Difference

The common difference is the backbone of an arithmetic sequence. It's the consistent value that separates each term from the next, creating a steady pattern. In simpler terms, the common difference \( d \) shows how much you need to add (or subtract) to go from one term to the next term in the sequence.
Here’s how the concept plays out:

• If the sequence is increasing, \( d \) is a positive number.
• If the sequence is decreasing, \( d \) is negative.
• The same value of \( d \) is applied uniformly across the sequence.

Understanding the common difference is a key part of working with arithmetic sequences. It's what allows you to predict the sequence of numbers. Let's say you're asked for the next term after a list of numbers like 3, 7, 11, 15,...: the next term will be 19. Why? Because as you can see, the common difference \( d \) is 4. This knowledge not only empowers you to compute terms but also to ensure the sequence's integrity and coherence.