Problem 48 - Precalculus with Limits

Question

In Exercises 47 - 50, the first two terms of the arithmetic sequence are given. Find the missing term. \( a_1 = 3, a_2 = 13, a_9 = \)

Step-by-Step Solution

Answer

The missing 9th term in the arithmetic sequence is 83.

Step 1: Calculate the common difference

Firstly, find the common difference (d) between the terms in the arithmetic sequence by subtracting the first term from the second term: \[ d = a_2 - a_1 = 13 - 3 = 10 \]

Step 2: Find the 9th term

Now, use the formula of the arithmetic sequence to find the 9th term: \[ a_9 = a_1 + (9 - 1) \cdot d = 3 + 8 \cdot 10 = 83 \]

Key Concepts

Common Difference

In arithmetic sequences, the 'common difference' is a fundamental component. This difference is constant between any two consecutive terms of the sequence. Let's see how it works with our example.

If you're looking at the sequence where the first term is 3 (\(a_1 = 3\)) and the second term is 13 (\(a_2 = 13\)), you find the common difference by subtracting the first term from the second: \[d = a_2 - a_1 = 13 - 3 = 10\].

This value of 10 is what you add to each term to get the next term in the sequence. So, starting with 3, if you keep adding 10, you get 13, then 23, 33, and so on. Remember: the common difference can be positive, negative, or zero, affecting how the sequence increases, decreases, or remains constant.

Arithmetic Sequence Formula

The arithmetic sequence formula is your best friend when you're solving problems related to sequences! This formula helps you find any term of the sequence if you know the first term and the common difference.

The general form of an arithmetic sequence is given by:\[a_n = a_1 + (n - 1) \cdot d\].

This formula lets you calculate the value of a term (\(a_n\)) when you know the first term (\(a_1\)), the position of the term (\(n\)), and the common difference (\(d\)).

For instance, if \(a_1\) = 3 and \(d\) = 10, this formula tells you the value of \(a_9\) will be \(a_1 + 8 \cdot 10\). This method is reliable and quick, so always keep it in mind for arithmetic sequences.

Finding Terms in a Sequence

Finding terms in an arithmetic sequence is pretty straightforward once you have the formula and the common difference. Let's break it down using an example for clear understanding.

If you're asked to find the ninth term (\(a_9\)) of our sequence where \(a_1 = 3\) and \(d = 10\), start by substituting these into the arithmetic sequence formula:\[a_9 = a_1 + (9 - 1) \cdot d\].

Plugging in the numbers, it simplifies to:\[a_9 = 3 + 8 \cdot 10 = 83\].

By following these steps, you can find any term in an arithmetic sequence. Just identify the common difference and use the sequence formula. This is the systematic way to solve for any term, whether near or far from the first.