Problem 28 - Algebra 2

Question

Find the missing term of each arithmetic sequence. \(.65, \square,-60, \dots\)

Step-by-Step Solution

Answer

The missing second term in the arithmetic sequence is -29.675

Step 1: Find the common difference

To find the common difference 'd', subtract the first term from the third term and divide it by 2 (because the missing term is the second term). So, \(d = \frac{-60 - .65}{2}\)

Step 2: Evaluate the common difference

Simplify the value of 'd'. Therefore, \(d = -30.325\)

Step 3: Determine the second term

The formula for the \(n^{th}\) term of an arithmetic sequence is \(a + (n-1)d\). Substitute 'a' with .65, 'd' with -30.325 and 'n' with 2 to find the second term. So, \(second \ term = .65 + (2-1)(-30.325)\)

Step 4: Evaluate the second term

Calculate the answer, \(second \ term = -29.675\)

Key Concepts

Common Difference

In arithmetic sequences, the "common difference" is the amount by which consecutive terms in the sequence increase or decrease. It's crucial because it tells us how the sequence progresses. Here’s how you calculate it:

• Pick two consecutive terms from the sequence. Let's call them the first term and the third term.
• Subtract the first term from the third term. This gives you the total change between these positions.
• Divide by the number of intervals between them. For example, if there’s one missing term in between, you’d divide by 2.

This gives you the average change per step, which is your common difference. In our example, by subtracting 0.65 from -60 and dividing by 2, we find the common difference to be -30.325. Knowing this difference helps us easily find any term in the sequence.

Nth Term Formula

The "nth term formula" for an arithmetic sequence is a handy mathematical tool. It allows you to calculate the value of any term in the sequence without needing to list all the preceding terms manually. The formula is:
\[ a_n = a_1 + (n-1) imes d \]
Where:

• \(a_n\): the value of the nth term.
• \(a_1\): the first term of the sequence.
• \(d\): the common difference.
• \(n\): the position of the term within the sequence.

To use the formula effectively, plug in the known values for the first term, the common difference, and the desired position (n).
For instance, in the example with the sequence starting at 0.65, to find the second term, we use the formula with n=2, resulting in -29.675.

Sequence Problems

Solving sequence problems requires a systematic approach, especially with arithmetic sequences. You will usually start by identifying the pattern or rule that defines the sequence. Here’s a structured approach:

• First, determine if the sequence is arithmetic by checking if the difference between consecutive terms is constant.
• Once confirmed, find the common difference, which is the key to unlocking further terms in the sequence.
• Next, apply the nth term formula if you need to find a specific term or missing terms in a sequence.

These steps allow you to quickly and accurately solve problems related to arithmetic sequences. In our exercise, identifying the pattern and using the nth term formula helped us find the missing term robustly. Practice with these steps improves your ability to tackle various sequence-related challenges efficiently.