Problem 4 - Algebra and Trigonometry

Question

True or false? If we know the first and second terms of an arithmetic sequence, then we can find any other term.

Step-by-Step Solution

Answer

True, we can find any term given the first and second terms of an arithmetic sequence.

Step 1: Understanding an Arithmetic Sequence

An arithmetic sequence is a sequence of numbers where the difference between consecutive terms, called the common difference, is constant. If the first term is \(a_1\) and the second term is \(a_2\), then the common difference \(d\) can be found by \(d = a_2 - a_1\).

Step 2: General Formula for an Arithmetic Sequence

The \(n^{th}\) term of an arithmetic sequence can be calculated using the formula \(a_n = a_1 + (n-1) \cdot d\), where \(a_1\) is the first term, \(n\) is the term number, and \(d\) is the common difference.

Step 3: Applying Given Information

Given that we know \(a_1\) and \(a_2\), we can determine \(d\). With \(a_1\) and \(d\), we can substitute these into the formula for \(a_n\) to find any term in the sequence.

Step 4: Conclusion on the Possibility of Finding Other Terms

Since we can deduce the common difference \(d\) and have the formula to determine any term \(a_n\) from \(a_1\) and \(d\), it is indeed possible to find any other term in the sequence from knowing just the first and second terms.

Key Concepts

Common Difference

The common difference in an arithmetic sequence is one of its most vital characteristics. It represents the constant amount added or subtracted to reach the next term in the sequence. Understanding this concept makes it easy to work with arithmetic sequences.
Let's start by considering the first and second terms of an arithmetic sequence:

• Let the first term be denoted as \( a_1 \).
• The second term will be \( a_2 \).

To find the common difference \( d \), simply subtract the first term from the second term: \( d = a_2 - a_1 \).
This difference \( d \) is used throughout the sequence to generate new terms. All terms are spaced evenly apart by this amount. For example:

• If \( a_1 = 3 \) and \( a_2 = 7 \), then \( d = 7 - 3 = 4 \).
• Every term thereafter increases by 4.

By knowing the common difference, you can effectively predict every other term in the sequence.

nth Term Formula

To pinpoint any term within an arithmetic sequence, the nth term formula is essential. This formula provides a method to calculate the value of any term when you know the first term and the common difference.
The formula for the nth term is:\[ a_n = a_1 + (n-1) \cdot d \]
Here's how it works:

• \( a_n \) is the term you're trying to find.
• \( a_1 \) is the first term of the sequence.
• \( n \) is the term number you are interested in.
• \( d \) is the common difference.

To use this formula, multiply the common difference \( d \) by \( (n-1) \), then add this result to the first term \( a_1 \). For example, if the common difference \( d \) is 2, and the first term \( a_1 \) is 5, to find the 4th term:

• Plug into the formula: \( a_4 = 5 + (4-1) \cdot 2 \)
• Simplify: \( a_4 = 5 + 6 = 11 \)

Understanding this formula lets you jump to any term in a sequence with ease.

Sequence Terms

Terms in an arithmetic sequence are elements or numbers arranged in order such that each term after the first is defined by adding the common difference to the previous term.
These terms follow a simple and predictable pattern, thanks to their relationship with the common difference. Here's what you need to know:

• The first term is often given, and subsequent terms are determined using the nth term formula.
• The second term can be calculated as the first term plus the common difference \( a_2 = a_1 + d \).

Each term can be written in terms of the first term \( a_1 \) and the common difference \( d \) as follows:

• Third term: \( a_3 = a_1 + 2d \)
• Fourth term: \( a_4 = a_1 + 3d \)
• And so on.

By understanding this pattern, you can easily determine any term in the sequence and appreciate the structured beauty of arithmetic sequences.