Problem 50 - Beginning and Intermediate Algebra

Question

Find the sum of the first 11 terms of the arithmetic sequence with first term \(-8\) and last term \(-58\).

Step-by-Step Solution

Answer

The sum of the first 11 terms of the arithmetic sequence with first term -8 and last term -58 is -363.

Step 1: Find the common difference

In an arithmetic sequence, the difference between consecutive terms remains constant. To find the common difference (denoted as d), we'll use the formula: \[ a_n = a_1 + (n-1)d \] where \(a_n\) is the nth term, \(a_1\) is the first term, n is the number of terms, and d is the common difference. We know the first term \(a_1 = -8\) and the last term \(a_{11} = -58\) (since there are 11 terms in the sequence), and we need to find d. We'll plug in the given values to the formula above to find d: \[ -58 = -8 + (11 - 1)d \]

Step 2: Solve for common difference d

Now we'll solve for d: \begin{align*} -58 &= -8 + 10d \\ -50 &= 10d \\ d &= -5 \end{align*} So the common difference between the consecutive terms of the sequence is -5.

Step 3: Use the arithmetic sequence sum formula

Now that we have the common difference, we can use the arithmetic sequence sum formula to find the sum of the first 11 terms. The sum of an arithmetic sequence is given by the formula: \[ S_n = \frac{n(a_1 + a_n)}{2} \] where \(S_n\) is the sum of n terms, n is the number of terms, \(a_1\) is the first term, and \(a_n\) is the nth term. We'll plug in the given values and the value of d that we just found into this formula: \[ S_{11} = \frac{11(-8 + (-58))}{2} \]

Step 4: Calculate the sum

Let's calculate the sum: \begin{align*} S_{11} &= \frac{11(-66)}{2} \\ &= \frac{-726}{2} \\ &= -363 \end{align*} So, the sum of the first 11 terms of the arithmetic sequence is -363.

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is the constant gap or difference between consecutive terms. It's what makes an arithmetic sequence linear and predictable. For instance, consider a sequence: -8, -13, -18, -23, and so forth. Here, if you subtract two successive terms, say \(-13 - (-8)\), you'll get \(-5\). This number, -5, is the common difference \(d\).
To find the common difference in a sequence, you can use the formula for the nth term of an arithmetic sequence, given by \(a_n = a_1 + (n-1)d\).
- Where \(a_n\) refers to the nth term.- \(a_1\) is the first term.- \(n\) is the number of terms.- \(d\) is the common difference. Understanding the common difference helps define the behavior of the entire sequence.

Sum of Arithmetic Sequence

Calculating the sum of an arithmetic sequence allows you to find the total when adding multiple terms together. The sum is found using a special formula:- \( S_n = \frac{n(a_1 + a_n)}{2} \)Here's how it works:
- \(S_n\) stands for the sum of the sequence.- \(n\) is the total number of terms you're adding.- \(a_1\) is the first term.- \(a_n\) is the last or nth term.
This formula effectively calculates the average of the first and last terms, then multiplies them by the number of terms to reach the total sum. Understanding this formula is crucial, as it simplifies the sometimes-complex task of summing up a long list of numbers.

First Term

The first term, typically denoted as \(a_1\), is the starting point of any sequence. It's the number from which the sequence begins to grow or decrease, based on the common difference. For example, in the sequence given in the exercise, the first term is \(-8\).
The importance of the first term extends beyond merely being the initial value; it significantly influences the entire sequence's structure. In our exercise, starting at \(-8\) and with a common difference of \(-5\) means the sequence extends in a decreasing order, adding depth to our understanding of sequences.
Having the first term helps in determining all other terms in the sequence. When combined with the nth term formula, it provides a roadmap for finding any specific term within the sequence. Additionally, it is crucial for using the sum of arithmetic sequence formula.

Nth Term Formula

The nth term formula is a handy tool that gives you the ability to pinpoint any particular term within an arithmetic sequence. This formula is expressed as:- \( a_n = a_1 + (n-1)d \)Here's a breakdown:- \(a_n\) is the term you're trying to find.- \(a_1\) again is the first term.- \((n-1)\) refers to the number of gaps between the first term and the term you wish to find.- \(d\) is the common difference.By plugging values into this formula, you can discover any term in the sequence just by knowing the first term and the common difference.
For example, if you wanted to find the 11th term in a sequence that starts with \(-8\) and has a common difference of \(-5\), you’d calculate: \( a_{11} = -8 + (11-1)(-5) \). Thus, not only does this formula provide the means to locate terms, but it also reinforces the structure and predictability inherent in arithmetic sequences.