1. What is an Arithmetic Sequence?

An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This difference is called the common difference (d). The nth term of an arithmetic sequence can be calculated using the formula:

2. How Does the Calculator Work?

The calculator uses the arithmetic sequence formula:

Where:

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

Explanation: The formula calculates any term in the sequence by starting with the first term and adding the common difference multiplied by one less than the term number.

3. Importance of Sequence Calculation

Details: Understanding arithmetic sequences is fundamental in mathematics and has applications in computer science, physics, finance, and many other fields where patterns of constant change occur.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What if the common difference is negative?
A: The sequence will decrease by that amount each term. The formula works the same way for positive and negative differences.

Q2: Can n be a decimal?
A: Typically n is a positive integer, but the formula mathematically works for any real number n.

Q3: What's the difference between sequence and series?
A: A sequence is an ordered list of numbers, while a series is the sum of the terms of a sequence.

Q4: Can this formula be used for geometric sequences?
A: No, geometric sequences use multiplication rather than addition between terms and have a different formula.

Q5: How do I find the sum of the first n terms?
A: The sum Sₙ of the first n terms is given by: \( S_n = \frac{n}{2}(2a_1 + (n-1)d) \) or \( S_n = \frac{n}{2}(a_1 + a_n) \)