Problem 21 - Thomas Calculus

Question

Find a formula for the \(n\)th term of the sequence. $$ 1,5,9,13,17, \dots $$

Step-by-Step Solution

Answer

The formula for the nth term is \(a_n = 4n - 3\).

Step 1: Identify Pattern

Look for a pattern in the sequence. Notice that each term increases by 4 from the previous term: \(5 - 1 = 4\), \(9 - 5 = 4\), and so on.

Step 2: Determine the Structure of Formula

Recognize that the sequence is arithmetic, where each term is increased by 4. Use the formula for the nth term of an arithmetic sequence: \(a_n = a_1 + (n-1) imes d\).

Step 3: Apply Known Values

Insert the known values into the formula: - The first term \(a_1 = 1\) - The common difference \(d = 4\) The formula becomes:\[a_n = 1 + (n-1) imes 4\].

Step 4: Simplify the Expression

Simplify the formula by expanding and combining like terms: \[a_n = 1 + 4n - 4\]\[a_n = 4n - 3\].

Key Concepts

Pattern Recognition in Sequences

In the world of sequences, identifying patterns is your first and most crucial step. Here, it's like being a detective who spots a common theme or trend among the numbers. In our example sequence: 1, 5, 9, 13, 17, and so forth, ask yourself if each number relates to the next in some consistent way.

• Look at the differences between consecutive terms: 5 minus 1, 9 minus 5, 13 minus 9, and 17 minus 13. Each of these calculations yields a difference of 4.
• This consistent increase suggests that we have a regular pattern.

Recognizing that a sequence increases by a constant number each time tells us we are dealing with an arithmetic sequence. This type of sequence has a common difference between each term, which is key to unlocking the "secret formula" for finding any term in the sequence.

nth Term Formula

Once you've spotted the pattern, it's time to express it with a formula. This special formula exists for arithmetic sequences, where each step forward involves adding a fixed amount, our common difference.Let's talk about the general formula for the nth term: \[ a_n = a_1 + (n-1) \times d \]

• Here, \(a_n\) represents the term you want to find.
• \(a_1\) is the first term in the sequence; in our example, it’s 1.
• \(d\) is the common difference, which is 4 in this sequence.
• \(n\) stands for the term number you are looking to calculate.

By identifying these parts, we can substitute them into the formula to find any term in the sequence.

Sequence Simplification

Now, onto sequence simplification: the art of making your life a whole lot easier. This is all about taking the formula we wrote and making it even simpler to use.Here's our formula in raw form: \[ a_n = 1 + (n-1) \times 4 \]Breaking it down into simpler terms turns math into magic because it becomes more accessible and user-friendly. To simplify, distribute the 4 into the parentheses:

• Multiply \((n-1)\) by 4: \( 4n - 4 \)
• Add the result to 1: \( 1 + 4n - 4 \)

Finally, combine like terms for a super smooth equation: \[ a_n = 4n - 3 \]This straightforward result means you can plug in any number for \(n\) and instantly find the corresponding term in the sequence, cutting the complexity for quick and easy results.