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Arithmetic Sequence Formula:
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An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is called the "common difference."
The calculator uses the arithmetic sequence formula:
Where:
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 position.
Details: Arithmetic sequences are used in various fields including finance (calculating simple interest), physics (uniform motion), computer science (algorithm analysis), and everyday situations like calculating seating arrangements or savings patterns.
Tips: Enter the first term of the sequence, the common difference between terms, and the position of the term you want to find. All values must be valid numbers with n being a positive integer.
Q1: What if the common difference is negative?
A: A negative common difference means the sequence is decreasing. The formula works the same way regardless of whether d is positive or negative.
Q2: Can n be a decimal or fraction?
A: No, n must be a positive integer as it represents the position of a term in the sequence (1st, 2nd, 3rd, etc.).
Q3: 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.
Q4: How do I find the sum of an arithmetic sequence?
A: The sum of the first n terms can be calculated using the formula: \( S_n = \frac{n}{2}(a_1 + a_n) \) or \( S_n = \frac{n}{2}[2a_1 + (n-1)d] \).
Q5: Can this formula be used for non-integer sequences?
A: Yes, the first term and common difference can be any real numbers (integers, fractions, decimals), creating sequences with non-integer terms.