Problem 44 - Contemporary Precalculus

Question

Find the kth partial sum of the arithmetic sequence \(\left\\{a_{n}\right\\}\) with common difference d. $$k=9, a_{1}=-4, d=\frac{1}{2}$$

Step-by-Step Solution

Answer

Answer: The 9th partial sum of the arithmetic sequence is -18.

Step 1: Find the formula for an arithmetic sequence

An arithmetic sequence can be represented as: $$a_n = a_1 + (n - 1)d$$ Where \(a_n\) is the nth term, \(a_1\) is the first term, \(n\) is the position in the sequence, and \(d\) is the common difference. In our case, \(a_1 = -4\) and \(d = \frac{1}{2}\).

Step 2: Find the 9th term in the sequence

Using the formula for the arithmetic sequence, we can find the 9th term by substituting in the values we know: $$a_9 = a_1 + (9 - 1)d$$ $$a_9 = -4 + (8)(\frac{1}{2})$$ $$a_9 = -4 + 4$$ $$a_9 = 0$$

Step 3: Find the sum of the arithmetic sequence

The sum of an arithmetic sequence can be found using the formula: $$S_k = \frac{k(a_1 + a_k)}{2}$$ Where \(S_k\) is the kth partial sum, in our case k = 9. Substituting the values we know into the formula, we get: $$S_9 = \frac{9(-4 + 0)}{2}$$ $$S_9 = \frac{9(-4)}{2}$$ $$S_9 = -18$$

Step 4: The final answer

The 9th partial sum of the arithmetic sequence with \(a_1=-4\) and a common difference of \(d=\frac{1}{2}\) is \(-18\).

Key Concepts

Partial Sum

When dealing with arithmetic sequences, the **partial sum** is the sum of the first \( k \) terms. You use the partial sum formula:

• \( S_k = \frac{k(a_1 + a_k)}{2} \)

In this formula, \( S_k \) stands for the partial sum, \( k \) is the number of terms you are summing, \( a_1 \) is the first term, and \( a_k \) is the \( k \)-th term.
Finding the partial sum can help in understanding how large or small the sum of the sequence grows as you add more terms.
For instance, if you have a sequence where the first term is -4, and you're adding up to the ninth term where each step increases by 0.5, substituting these into the formula can help determine the sum of these terms.

Common Difference

Arithmetic sequences rely heavily on what's known as the **common difference**. It's the amount each term increases or decreases compared to the previous one.

• The general formula for the nth term is \( a_n = a_1 + (n-1)d \), where \( d \) is the common difference.
• To find any term in the sequence, you add the common difference multiplied by the step number minus one, to the first term.

Understanding this concept is crucial when you're making calculations within arithmetic sequences.
Given a common difference of \( \frac{1}{2} \), you know each successive number in the sequence will increase by that fixed amount.

Sequence Formula

The **sequence formula** is indispensable for identifying each term in an arithmetic sequence. It's how you find any particular term without listing each previous one:

• \( a_n = a_1 + (n-1)d \)

This formula makes life easier, streamlining the process of finding specific terms in arithmetic patterns.
Here, \( a_1 \) is the starting point, \( n \) is the desired term’s position, and \( d \) is the common difference.
For instance, if you're tasked with finding the ninth term and your sequence starts at -4 with a common difference of \( \frac{1}{2} \), this formula allows you to quickly get the result, which is 0 in this case.
This sequential finding is crucial for moving forward in understanding larger structures and sums in a series.