Problem 24 - Precalculus: Mathematical for Calculus

Question

Determine the common difference, the fifth term, the \(n\) th term, and the 100 th term of the arithmetic sequence. $$1,5,9,13, \dots$$

Step-by-Step Solution

Answer

Common difference is 4, fifth term is 17, nth term is \(1+(n-1)\times4\), and 100th term is 397.

Step 1: Identify the First Term

The first term of the sequence given is 1. We denote this by \(a_1 = 1\).

Step 2: Calculate the Common Difference

An arithmetic sequence has a common difference (\(d\)) between consecutive terms. To find \(d\), subtract the first term from the second term: \(d = 5 - 1 = 4\).

Step 3: Write the Formula for the nth Term

The \(n\)th term of an arithmetic sequence is given by the formula \(a_n = a_1 + (n-1)d\). Here, \(a_1 = 1\) and \(d = 4\), so the formula becomes \(a_n = 1 + (n-1) \times 4\).

Step 4: Calculate the Fifth Term

Substitute \(n = 5\) into the \(n\)th term formula: \(a_5 = 1 + (5-1) \times 4 = 1 + 16 = 17\).

Step 5: Calculate the 100th Term

Substitute \(n = 100\) into the \(n\)th term formula: \(a_{100} = 1 + (100-1) \times 4 = 1 + 396 = 397\).

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is a crucial element that defines how we move from one term to the next. Think of it as the consistent amount that you add (or subtract, if the difference is negative) to a term to get to the subsequent term. This makes arithmetic sequences very predictable and easy to work with.

To find this common difference, simply take any term in the sequence and subtract the previous term. For example, in the sequence given: 1, 5, 9, 13, ..., you subtract the first term from the second term:

• 5 - 1 = 4

The common difference here is 4. That's like taking steps on a ladder where each step is 4 units up from the last one. With this regular step size, you can calculate any term in the sequence once you know the starting point and the number of steps.

Nth Term Formula

The Nth term formula is your key to finding any term in an arithmetic sequence without having to list all terms up to that point. Once you have the common difference and the first term, you can easily find the Nth term using the formula. This formula for an arithmetic sequence is:

• \(a_n = a_1 + (n-1) \times d\)

Here, \(a_n\) represents the Nth term you want to find, \(a_1\) is the first term of the sequence, \(n\) is the term number, and \(d\) is the common difference.

This formula allows you to "jump" directly to any term in the sequence. For the given sequence: 1, 5, 9, 13, using \(a_1 = 1\) and \(d = 4\), the formula becomes \(a_n = 1 + (n-1) \times 4\). Plug in the desired \(n\) value, and you instantly have the term's value. For instance, seeking the 5th term is as simple as plugging 5 for \(n\) into the equation, yielding \(a_5 = 1 + (5-1) \times 4 = 17\). This calculation is not only quick but also highly practical for large \(n\) values.

Arithmetic Progression

An arithmetic progression is a sequence of numbers where the difference between any two successive terms is constant. This structure defines a straight path through the number sequence, which is very systematic and predictable.

In every arithmetic progression, knowing the first term and the common difference lets you build the entire sequence. You begin with the initial term and apply the common difference to determine each subsequent term. The sequence given in the exercise, 1, 5, 9, 13, ..., is a perfect example. Starting at 1 and repeatedly adding 4 constructs the sequence.

This predictability means arithmetic progressions are not only straightforward but powerful. They appear in many areas of mathematics and day-to-day situations, making them an essential concept to master. Whether it's finding the 100th term or just understanding the pattern, grasping the concept of an arithmetic progression boasts significant utility.