Problem 28 - 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. $$a_{1}=-6, d=\frac{2}{3}$$

Step-by-Step Solution

Answer

Answer: The fifth term of the arithmetic sequence is -10/3 and the formula for the nth term is a_n = -6 + (2(n - 1))/3.

Step 1: Identify given information

We have the following information: First term, \(a_1 = -6\) Common difference, \(d = \frac{2}{3}\)

Step 2: Find the fifth term (\(a_5\))

We will use the arithmetic sequence formula: \(a_n = a_1 + (n - 1)d\) To find the fifth term, let \(n=5\): \(a_5 = a_1 + (5 - 1)d\) \(a_5 = -6 + (4)\left(\frac{2}{3}\right)\) \(a_5 = -6 + \frac{8}{3}\) \(a_5 = \frac{-10}{3}\) So the fifth term of the sequence is \(a_5 = \frac{-10}{3}\).

Step 3: Find the formula for the nth term

Now let's generalize the arithmetic sequence formula using the given values of \(a_1\) and \(d\): \(a_n = a_1 + (n - 1)d\) \(a_n = -6 + (n - 1)\left(\frac{2}{3}\right)\) \(a_n = -6 + \frac{2(n - 1)}{3}\) This is the formula for the nth term of the arithmetic sequence with \(a_1 = -6\) and \(d = \frac{2}{3}\). To summarize, the fifth term is \(a_5 = \frac{-10}{3}\) and the formula for the nth term is \(a_n = -6 + \frac{2(n - 1)}{3}\).

Key Concepts

nth term formula

The nth term formula is a vital tool when dealing with arithmetic sequences. It allows you to find any term in the sequence without having to list out all the previous terms. In arithmetic sequences, the nth term is expressed as: \[ a_n = a_1 + (n - 1)d \] where:

• \(a_n\) is the nth term you're looking to find.
• \(a_1\) is the first term of the sequence.
• \(d\) is the common difference, which describes how much each term increases (or decreases) from the previous one.
• \(n\) is the term number.

This formula is powerful and concise. It helps you calculate any term's value by plugging in the respective term number and performing simple arithmetic operations. Keep this formula handy when working with arithmetic sequences, as it simplifies finding terms greatly.

common difference

The common difference in an arithmetic sequence is a constant value that you add (or subtract) to each term to get to the next term. This difference is crucial because it determines the pattern and rate of increase or decrease in the sequence. To find the common difference, you can subtract any term from the one following it. For instance, if the sequence is 2, 5, 8, 11, the common difference \(d\) is: \[ d = 5 - 2 = 3 \] So, each term increases by 3. This difference is uniform throughout the sequence and serves as a foundational element in forming the sequence's structure. Knowing the common difference helps not only in forming the sequence but also in predicting future terms using the nth term formula.

first term of sequence

The first term in an arithmetic sequence is key to defining the sequence. Represented as \(a_1\), this term is the starting point from which all other terms are derived by adding the common difference repeatedly. For example, if the first term \(a_1\) is -6 and the common difference \(d\) is \(\frac{2}{3}\), then the sequence starts like this: -6, -6 + \(\frac{2}{3}\), -6 + 2*\(\frac{2}{3}\), and so on. The value of \(a_1\) sets the sequence into motion, establishing its base value. Remember, accurately identifying \(a_1\) is as important as knowing the common difference and the nth term formula, as it directly affects the calculation and correctness of every subsequent term in the sequence.