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Arithmetic Sequence Formula:
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An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This difference is called the common difference (d). Each term after the first is found by adding the common difference to the preceding term.
The calculator uses the arithmetic sequence formula:
Where:
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.
Details: Understanding arithmetic sequences is fundamental in mathematics and has applications in finance, computer science, physics, and many other fields where patterns of constant change occur.
Tips: Enter the first term of your sequence, the term number you want to find, and the common difference between terms. All values must be valid numbers.
Q1: What's the difference between arithmetic and geometric sequences?
A: In arithmetic sequences, the difference between terms is constant (addition). In geometric sequences, the ratio between terms is constant (multiplication).
Q2: Can n be a decimal or fraction?
A: In standard arithmetic sequences, n must be a positive integer as it represents the term position in the sequence.
Q3: What if the common difference is negative?
A: A negative common difference means each term is smaller than the previous one, creating a decreasing sequence.
Q4: How is this used in real life?
A: Arithmetic sequences model situations with constant change like weekly savings, depreciation, or simple interest.
Q5: What's the formula for the sum of an arithmetic sequence?
A: The sum of the first n terms is \( S_n = \frac{n}{2} \times (2a_1 + (n - 1)d) \) or \( S_n = \frac{n}{2} \times (a_1 + a_n) \).