Problem 44 - Beginning and Intermediate Algebra

Question

Find the number of terms in each arithmetic sequence. $$-7,-11,-15,-19, \ldots,-91$$

Step-by-Step Solution

Answer

There are 22 terms in the arithmetic sequence.

Step 1: Identify the first term, last term, and common difference

The first term, \(a_1\), is -7, the last term, \(a_n\), is -91, and the common difference, d, is -4.

Step 2: Use the formula of an arithmetic sequence to find the number of terms

We will now use the formula to find n: $$-91 = -7 + (n - 1)(-4)$$

Step 3: Solve for n

Now, we will solve the equation for n: \begin{align*} -91 &= -7 + (n - 1)(-4) \\ -84 &= (n - 1)(-4) \\ 21 &= n - 1 \\ n &= 22 \end{align*} So, there are 22 terms in this arithmetic sequence.

Key Concepts

Common Difference

In the realm of arithmetic sequences, the 'common difference' is a key concept. It refers to the difference between each consecutive term in the sequence.

For instance, in the given sequence,

• First term: -7
• Second term: -11
• Third term: -15
• and so on…

The difference between each of these terms is \(-11 - (-7) = -4\) or \(-15 - (-11) = -4\). This difference once identified, remains constant throughout the sequence.

Understanding the common difference helps you easily predict future terms and solve many sequence-related problems. It becomes a simple subtraction task, reducing the complexity of analyzing sequences.

Number of Terms in a Sequence

Finding the number of terms in an arithmetic sequence is essential to solving problems within it. In our given sequence from -7 to -91, this is what was solved.

The concept is straightforward; you start with your first term and move to your final term by repeatedly adding the common difference. But how do you find out how many times this addition occurs?

By using the sequence formula, anecdotal in the exercise, you rearrange it to solve for 'n', representing the number of terms:

• Recognize your first term, last term, and common difference.
• Plug them into the equation and solve for 'n'.

This gives us the total count of digits forming the sequence from beginning to end, crucial for comprehensive sequence analysis.

Sequence Formula

The sequence formula plays an instrumental role in arithmetic sequences. The general formula for an arithmetic sequence is: \[ a_n = a_1 + (n-1) imes d \] where:

• \(a_n\) is the last term,
• \(a_1\) is the first term,
• \(n\) is the number of terms, and
• \(d\) is the common difference.

Using this formula, you can easily determine any term within the sequence given sufficient known information about the surrounding values.

The formula helps dissect the problem by organizing data like first term, last term, and difference into a manageable equation. Understanding this makes dealing with sequences systematic rather than chaotic, offering a reliable method to find values within a sequence.

Solving Equations in Sequences

Solving equations in sequences may seem daunting, but it's effective with a structured approach. It involves simplifying an equation to identify unknown elements such as the number of terms or a specific term's value.

In the exercise, we started with an equation that uses known values to find an unknown:\(-91 = -7 + (n - 1)(-4)\). By solving this, you:

• Align terms to simplify the equation.
• Isolate the variable 'n' (representing number of terms).
• Use basic algebra to find values.

This process unravels the sequence, translating it from opaque data to accessible figures. Once you're comfortable solving these kinds of equations, arithmetic sequences become much easier to handle, transforming them from mere numbers into well-explained scenarios.