Problem 47 - Contemporary Precalculus

Question

Find the kth partial sum of the arithmetic sequence \(\left\\{a_{n}\right\\}\) with common difference d. $$k=9, a_{1}=6, a_{9}=-24$$

Step-by-Step Solution

Answer

Answer: The 9th partial sum of the arithmetic sequence is -81.

Step 1: Find the common difference (d)

Using the following arithmetic sequence formula, we can find the common difference (d): $$a_n = a_1 + (n - 1)d$$ Substitute \(a_9 = -24\), \(a_1 = 6\), and \(n=9\) into the formula: $$-24 = 6 + (9 - 1)d$$ Solve for d: $$-30 = 8d$$ $$d = -\frac{30}{8} = -3.75$$

Step 2: Find the kth partial sum

We can find the kth partial sum using the following formula for the sum of an arithmetic sequence: $$S_k = \frac{k}{2}(a_1 + a_k)$$ Substitute \(k=9\), \(a_1=6\), and \(a_k=a_9=-24\) into the formula: $$S_9 = \frac{9}{2}(6 - 24)$$ Calculate the kth partial sum: $$S_9 = \frac{9}{2} \times (-18) = -81$$ The 9th partial sum of the arithmetic sequence is -81.

Key Concepts

Partial Sum

Understanding the concept of partial sums in arithmetic sequences is crucial for determining the sum of terms up to a certain point. A partial sum of an arithmetic sequence refers to the total of a specific number of terms starting from the first term in the sequence. The formula to calculate the partial sum \(S_k\) of the first \(k\) terms in an arithmetic sequence is given as:

• \(S_k = \frac{k}{2}(a_1 + a_k)\)

In this formula:

• \(k\) is the number of terms to sum
• \(a_1\) is the first term
• \(a_k\) is the \(k\)th term

Substituting the known values into this equation allows us to easily compute the desired sum. For instance, with \(k=9\), \(a_1=6\), and \(a_k=-24\), the partial sum is calculated to be \(-81\), as shown in the original solution. This formula is instrumental in simplifying the calculation by eliminating the need to individually add each term.

Common Difference

In any arithmetic sequence, the concept of a common difference is the backbone of its structure. The common difference, denoted by \(d\), is the fixed amount added to each term to arrive at the next term in the sequence. Recognizing how to find this difference is key in identifying any term position within the sequence. The formula used to uncover \(d\) in an arithmetic sequence is:

• \(a_n = a_1 + (n - 1)d\)

Here:

• \(a_n\) is the term of interest
• \(a_1\) is the initial term
• \(n\) is the position of the term

Substituting values into this formula, like \(a_9 = -24\), \(a_1 = 6\), and \(n=9\), allows one to solve for \(d\). In this example, the common difference is calculated to be \(-3.75\). This constant forms the predictable pattern of the series, enabling easier tracking and prediction of subsequent terms.

Arithmetic Sequence Formula

An arithmetic sequence is a sequence of numbers with a constant addition, known as the common difference, from one term to the next. The formula that represents this sequence is pivotal to figuring out any term within the sequence and understanding its progression. The general arithmetic sequence formula is:

• \(a_n = a_1 + (n - 1)d\)

Breaking this down:

• \(a_n\) is the \(n\)th term in the sequence
• \(a_1\) is the sequence's first term
• \(d\) is the common difference
• \(n\) is the position of the term

By using this formula, you can calculate any term if the first term and the common difference are known. For example, with \(a_1=6\) and \(d=-3.75\), finding any subsequent term is straightforward. The arithmetic sequence formula not only aids in constructing the sequence but is also critical for deeper calculations such as determining partial sums or the number of terms in sequences.