1. What is an Arithmetic Sequence?

An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is called the "common difference" and is denoted by 'd'.

2. How Does the Formula Work?

The arithmetic sequence formula is:

Where:

• \( a_n \) — The nth term of the sequence
• \( a_1 \) — The first term of the sequence
• \( d \) — The common difference between terms
• \( n \) — The position of the term in the sequence

Explanation: The formula calculates any term in an arithmetic sequence by starting with the first term and adding the common difference multiplied by one less than the term's position.

3. Applications of Arithmetic Sequences

Details: Arithmetic sequences are used in various real-world applications including financial calculations, physics problems, computer algorithms, and pattern recognition in mathematics.

4. Using the Calculator

Tips: Enter the first term of your sequence, the common difference between terms, and the position of the term you want to calculate. All values must be valid numbers.

5. Frequently Asked Questions (FAQ)

Q1: What if the common difference is negative?
A: A negative common difference means the sequence is decreasing. The formula works exactly the same way with negative values.

Q2: Can this formula be used for non-integer terms?
A: Yes, the arithmetic sequence formula works with any real numbers, including fractions and decimals.

Q3: How do I find the sum of an arithmetic sequence?
A: The sum of the first n terms is given by \( S_n = \frac{n}{2}(a_1 + a_n) \) or \( S_n = \frac{n}{2}[2a_1 + (n-1)d] \).

Q4: What's the difference between arithmetic and geometric sequences?
A: In arithmetic sequences, the difference between terms is constant. In geometric sequences, the ratio between terms is constant.

Q5: Can n be zero or negative?
A: In standard arithmetic sequences, n represents the position in the sequence and should be a positive integer (1, 2, 3, ...).