Problem 43 - Contemporary Precalculus

Question

Find the kth partial sum of the arithmetic sequence \(\left\\{a_{n}\right\\}\) with common difference d. $$k=7, a_{1}=\frac{3}{4}, d=-\frac{1}{2}$$

Step-by-Step Solution

Answer

Answer: The 7th partial sum of the arithmetic sequence is -21/4.

Step 1: Replace the parameters in the equation

Plug in the given values of k, \(a_1\), and d into the formula: $$S_7 = \frac{7}{2} \left(2\cdot\frac{3}{4} + (7 - 1)\left(-\frac{1}{2}\right)\right)$$

Step 2: Evaluate the expression in parenthesis

Calculate the expression inside the parenthesis: $$2\cdot\frac{3}{4} = \frac{6}{4} = \frac{3}{2}$$ $$(7 - 1)\left(-\frac{1}{2}\right) = 6\left(-\frac{1}{2}\right) = -3$$ Now the expression is: $$S_7 = \frac{7}{2} \left(\frac{3}{2} + (-3)\right)$$

Step 3: Solve the expression inside the parenthesis

Add the numbers inside the parenthesis: $$\frac{3}{2} + (-3) = \frac{3}{2} - \frac{6}{2} = -\frac{3}{2}$$ Now the expression is: $$S_7 = \frac{7}{2} \cdot \left(-\frac{3}{2}\right)$$

Step 4: Multiply the two fractions

We need to multiply the two fractions together: $$\frac{7}{2} \cdot \left(-\frac{3}{2}\right) = -\frac{21}{4}$$

Step 5: Write the final answer

We found that the 7th partial sum of the arithmetic sequence is: $$S_7 = -\frac{21}{4}$$

Key Concepts

Partial Sum

The concept of a partial sum refers to the sum of the first few terms of a sequence, up to a specified term, denoted as "k". For example, the 7th partial sum of an arithmetic sequence is the sum of the first 7 terms of the sequence. To find a partial sum in an arithmetic sequence, you use the partial sum formula:

• The formula is: \[S_k = \frac{k}{2} (2a_1 + (k-1)d)\]where \(S_k\) is the k-th partial sum.
• Here, \(a_1\) is the first term, \(d\) is the common difference, and \(k\) is the number of terms you want to add together.

Finding a partial sum helps in determining the total accumulation of a sequence up to a certain point, which is crucial in various mathematical and real-life scenarios such as calculating total costs or specific payments over time.

Common Difference

In an arithmetic sequence, the common difference is the fixed amount each term increases or decreases by as you move from one term to the next. Knowing the common difference makes it easier to construct the whole sequence once you know the first term. For example, if the first term (\(a_1\)) is \(\frac{3}{4}\) and the common difference (d) is \(-\frac{1}{2}\), then you can find the next terms by continually subtracting \(\frac{1}{2}\) from each preceding term. This regular pattern is what characterizes arithmetic sequences and differentiates them from geometric sequences or other types of numeric patterns.

• To find the n-th term: \[a_n = a_1 + (n-1)d\]
• This formula reveals how sequences develop simply and predictably.

Thus, understanding the common difference is key to working with and understanding arithmetic sequences.

Sequence Formula

The sequence formula in arithmetic sequences enables you to find any term in the sequence without listing all the previous terms. This is crucial when you need to find terms that are deep into the sequence, such as the 100th or 1000th term. The standard formula for finding the n-th term is:

• \[a_n = a_1 + (n-1)d\]
• Here, \(a_n\) is the term you want to find.
• \(a_1\) is the first term, \(n\) is the term number, and \(d\) is the common difference.

This formula simplifies calculations and saves time, making it an essential tool for both academic studies and various applications in fields like finance and engineering, where predicting outcomes based on regular sequences is necessary.

Series Calculation

Calculating the total of a series means finding the sum of its terms, which is pivotal in many mathematical applications. In arithmetic sequences, the series calculation differs from simply knowing individual terms. The focus is on adding these terms to reach a total sum, which can be that's either finite or infinite, depending on the series length. In the context of our exercise:

• To calculate the series from the arithmetic sequence for the first 7 terms:\[S_7 = \frac{7}{2} (2 \cdot \frac{3}{4} + 6 \cdot ( - \frac{1}{2} ))\]
• This sum helps in understanding the bulk or cumulative quantity of the sequence.

Series calculations form the basis for tackling a range of problems, from basic mathematics to complex analyses in physics and economics, where determining accumulated change is pivotal.