Problem 32 - Contemporary Precalculus

Question

The first term \(a_{1}\) and the common difference d of an arithmetic sequence are given. Find the fifth term and the formula for the nth term. $$a_{1}=-.1, d=-8$$

Step-by-Step Solution

Answer

Answer: The formula for the nth term of the arithmetic sequence is $$a_n = -0.1 + (n-1) \times (-8)$$.

Step 1: Find the fifth term in the arithmetic sequence

To find the fifth term, we first need to know that the formula for the nth term in an arithmetic sequence is given by: $$a_{n} = a_{1} + (n-1) \times d$$ where \(a_{n}\) is the nth term, \(a_{1}\) is the first term, \(n\) is the position of the term, and \(d\) is the common difference. In this case, we want to find the fifth term, so we plug in \(a_{1} = -0.1\), \(n = 5\) and \(d = -8\) into the formula: $$a_{5} = -0.1 + (5-1) \times (-8)$$

Step 2: Compute the result

Now we just need to compute the result: $$a_{5} = -0.1 + (4) \times (-8)$$ $$a_{5} = -0.1 - 32$$ $$a_{5} = -32.1$$ So, the fifth term is \(-32.1\).

Step 3: Write the formula for the nth term

Now that we have the fifth term, we can use this information to write the formula for the nth term using the same formula we used before. With \(a_{1} = -0.1\), and \(d = -8\), the formula for the nth term is: $$a_{n} = -0.1 + (n-1) \times (-8)$$ So, the formula for the nth term is: $$ a_n = -0.1 + (n-1) \times (-8)$$

Key Concepts

nth term formula

An arithmetic sequence is a sequence of numbers where the difference between consecutive terms stays the same – this is the common difference, denoted by \(d\). To find any term in an arithmetic sequence, we use the nth term formula:\[a_{n} = a_1 + (n-1) \times d\]where:

• \(a_{n}\): the nth term you're looking to find
• \(a_1\): the first term in the sequence
• \(n\): the position of the term in the sequence
• \(d\): the common difference

Using this formula, you can plug in the values of \(a_1\), \(n\), and \(d\) to find any term in the sequence by calculating from the first term. This formula is especially useful because it saves the effort of calculating each term one by one, allowing instead for a direct jump to any designated position in the sequence.

common difference

The common difference in an arithmetic sequence is what makes this type of series unique. It's the number that is added (or subtracted if negative) to each term to get to the next one. In any given arithmetic sequence, the sequence is defined by just two key components: the first term (\(a_1\)) and the common difference (\(d\)).
For example, if the first term \(a_1\) is \(-0.1\) and the common difference \(d\) is \(-8\), each term is \(-8\) units less than the previous term. This repetitive pattern continues infinitely in either direction. Knowing the common difference allows us to easily predict any term or evaluate the change across any span of the sequence. The sign of the common difference impacts whether the sequence is increasing or decreasing. A negative \(d\) indicates a sequence that diminishes, as illustrated in our example.

fifth term

The fifth term in an arithmetic sequence can be found by using the nth term formula, specifically setting \(n = 5\). To find the fifth term, input the known values into the formula \[a_n = -0.1 + (n-1) \times (-8)\]
This breaks down into the calculation \[a_5 = -0.1 + 4 \times (-8)\]
Which simplifies to: \[a_5 = -0.1 - 32\]
Finally, we find that \(a_5 = -32.1\). This means the fifth term of this particular arithmetic sequence is \(-32.1\). By applying the formula, we quickly targeted the precise term we were seeking in the sequence without having to compute each of the preceding terms. Knowing how to find specific terms is essential in analyzing arithmetic sequences effectively.