Problem 44 - Algebra and Trigonometry

Question

39-44 Find the partial sum \(S_{n}\) of the arithmetic sequence that satisfies the given conditions. $$a_{2}=8, a_{5}=9.5, n=15$$

Step-by-Step Solution

Answer

The partial sum \( S_{15} \) is 165.

Step 1: Identify key values

The problem gives us two terms of the arithmetic sequence: \( a_2 = 8 \) and \( a_5 = 9.5 \), as well as \( n = 15 \) for the partial sum. First, let's denote the first term of the sequence by \( a_1 \) and the common difference by \( d \). We need to find these before we can find the partial sum.

Step 2: Set up equations for terms

Using the formula for the \(n\)-th term of an arithmetic sequence, \( a_n = a_1 + (n-1)d \), we can set up the following equations: \( a_2 = a_1 + d = 8 \) and \( a_5 = a_1 + 4d = 9.5 \).

Step 3: Solve for arithmetic sequence parameters

From the equations from the previous step, we substitute \( a_1 + d = 8 \) (equation 1) into \( a_1 + 4d = 9.5 \) (equation 2). Subtract equation 1 from equation 2 to solve for \( d \): \( 4d - d = 9.5 - 8 \), which simplifies to \( 3d = 1.5 \). Thus, \( d = 0.5 \).

Step 4: Find first term

Substitute \( d = 0.5 \) back into equation 1: \( a_1 + 0.5 = 8 \), which gives \( a_1 = 7.5 \). Now, we have the parameters: \( a_1 = 7.5 \) and \( d = 0.5 \).

Step 5: Calculate partial sum

The formula for the partial sum of an arithmetic sequence is \( S_n = \frac{n}{2} (2a_1 + (n-1)d) \). Substitute the known values (\( n = 15 \), \( a_1 = 7.5 \), \( d = 0.5 \)) into the formula: \[ S_{15} = \frac{15}{2} (2 \times 7.5 + (15-1) \times 0.5) \].

Step 6: Simplify and calculate

First calculate \( 2 \times 7.5 = 15 \). Then \( (15-1) \times 0.5 = 7 \). Therefore, \( S_{15} = \frac{15}{2} (15 + 7) = \frac{15}{2} \times 22 \). Finally, calculate \( \frac{15}{2} \times 22 = 165 \).

Key Concepts

Partial Sum

A partial sum in an arithmetic sequence represents the sum of a certain number of terms in the sequence. If you want to find the partial sum of the first 15 terms, denoted by \( S_{15} \), you’ll need the first term and the common difference. The formula used is:

• \( S_n = \frac{n}{2} (2a_1 + (n-1)d) \)
• Here, \( S_n \) is the partial sum, \( n \) is the number of terms, \( a_1 \) is the first term, and \( d \) is the common difference.

The idea is essentially to average the first and last terms we're considering, and then multiply by the number of terms to get the total. This method works well since arithmetic sequences have a consistent pattern determined by \( d \). Remember to plug in the correct values for \( n \), \( a_1 \), and \( d \) to find the sum quickly. Look at how these components help calculate without summing each term manually.

Common Difference

The common difference, denoted as \( d \), is a key element in arithmetic sequences, defining the consistent increase or decrease between consecutive terms. In the example given, after identifying terms like \( a_2 = 8 \) and \( a_5 = 9.5 \), you need to find \( d \). An arithmetic sequence progresses with this steady interval expressed by the equation:

• \( a_n = a_1 + (n-1)d \)

Here, by solving equations like \( a_5 = a_1 + 4d = 9.5 \) against \( a_2 = a_1 + d = 8 \), we discover \( d = 0.5 \). The ability to identify and calculate \( d \) enables proper construction and understanding of the sequence structure. Knowing \( d \) is crucial, as it affects how one calculates the sum of multiple terms (or partial sums), and how they establish new values within the sequence.

First Term of Sequence

The first term of an arithmetic sequence, often represented as \( a_1 \), serves as the starting point for creating and understanding the sequence. To find \( a_1 \), you rely on given terms of the sequence alongside the common difference. In our example, after determining \( d = 0.5 \), solving for \( a_1 \) using \( a_2 = a_1 + 0.5 = 8 \), you find \( a_1 = 7.5 \). This establishes the baseline for the sequence, around which all other terms will revolve, determined by the common difference.

• Once you know \( a_1 \) and \( d \), you gain insight into every subsequent term through the straightforward application of the arithmetic formula \( a_n = a_1 + (n-1)d \).

This clarity is fundamental in both listing terms and calculating sums within any arithmetic progression.

Arithmetic Sequence Formula

The arithmetic sequence formula is pivotal in defining and computing the elements of an arithmetic sequence. It is formally expressed as:

• \( a_n = a_1 + (n-1)d \)

This formula provides a mechanism to determine any term in the sequence by using the first term \( a_1 \) and the common difference \( d \). It's important because it lays the foundation for further analysis, like finding a specific term's value quickly without enumerating all previous terms.

• For instance, given \( a_1 = 7.5 \) and \( d = 0.5 \), to find the 5th term \( a_5 \): substitute into the formula to get \( a_5 = 7.5 + (5-1) \times 0.5 = 9.5 \).

This formula simplifies the understanding of sequence behavior and facilitates the solution of related problems, like partial sums or obtaining term values efficiently.