Sequence Calculator

What is a Sequence?

In mathematics, a sequence is an ordered list of objects, typically numbers, called terms. Sequences can be finite (having a specific number of terms) or infinite. Each term in the sequence is usually identified by its position or index, often denoted by the variable n.

Sequences are often defined by a rule or formula that specifies how to calculate the value of the n-th term (denoted as a_n) based on its index n. This is called an explicit formula.

Common types of sequences include:

• Arithmetic Sequences: Each term is found by adding a constant difference to the previous term (e.g., 2, 5, 8, 11,... where the formula is a_n = 2 + (n-1)*3 for n=1, 2, 3...).
• Geometric Sequences: Each term is found by multiplying the previous term by a constant ratio (e.g., 3, 6, 12, 24,... where the formula is a_n = 3 * 2^n-1 for n=1, 2, 3...).
• Many other sequences are defined by more complex formulas involving powers, trigonometric functions, logarithms, etc.

How This Sequence Calculator Works

This calculator generates terms of a sequence based on an explicit formula you provide. It also calculates the sum of the generated terms. Here’s the process:

1. Input Formula (a_n): Enter the explicit formula for the n-th term of the sequence in the first field. You must use 'n' as the index variable in your formula. You can use standard arithmetic operators (+, -, *, /, %) and JavaScript's built-in Math functions (e.g., pow(base, exponent) for powers, sin(n), cos(n), exp(n), log(n), sqrt(n), constants like PI).
2. Input Start Index (n₀): Specify the index value for the first term you want to calculate (e.g., 1 or 0).
3. Input Number of Terms: Enter how many consecutive terms of the sequence you wish to generate, starting from the Start Index.
4. Click Calculate: Press the "Generate Sequence" button.
5. Calculation Loop: The calculator iterates from n = Start Index up to (Start Index + Number of Terms - 1). In each iteration:
   • It evaluates the formula you provided using the current value of n to find the term a_n.
   • It keeps track of each calculated term and calculates the running sum.
   It uses JavaScript's function constructor to interpret your formula string.
6. Display Results: The calculator shows:
   • A list of the generated sequence terms.
   • The sum of all the generated terms.
   It includes error handling for invalid formulas or calculation issues.

Frequently Asked Questions (FAQs)

Q1: How do I enter the formula? What functions can I use?

Use 'n' as the variable representing the index. Use standard mathematical operators: +, -, * (multiplication), / (division), % (modulo/remainder). You can also use parentheses () for grouping.
For more advanced functions, use JavaScript's Math object methods:

• Powers: pow(base, exponent) (e.g., pow(n, 2) for n², pow(2, n) for 2ⁿ)
• Trigonometry: sin(n), cos(n), tan(n) (angles in radians)
• Exponentials/Logs: exp(n) (eⁿ), log(n) (natural log), log10(n) (base-10 log), log2(n) (base-2 log)
• Roots: sqrt(n) (square root), cbrt(n) (cube root)
• Rounding: round(n), floor(n), ceil(n)
• Absolute Value: abs(n)
• Constants: PI (π ≈ 3.14159), E (e ≈ 2.718)

Example: To represent a_n = 3n² - 5, enter 3 * pow(n, 2) - 5 or 3*n*n - 5.

Q2: Can I calculate terms for Arithmetic or Geometric sequences?

Yes, by entering their explicit formulas:

• Arithmetic: Formula is a_n = a₁ + (n - 1)d, where a₁ is the first term and d is the common difference. Example: For first term 5, difference 3, enter 5 + (n-1)*3 (assuming start index n=1).
• Geometric: Formula is a_n = a₁ * r^n-1, where a₁ is the first term and r is the common ratio. Example: For first term 2, ratio 4, enter 2 * pow(4, n-1) (assuming start index n=1).

Q3: Can the start index 'n' be zero?

Yes. You can set the "Start Index" field to 0 or any other integer. The calculator will then start evaluating the formula for n = 0, 1, 2,... for the specified number of terms.

Q4: Is there a limit to the number of terms I can generate?

Yes. To prevent performance issues or browser freezing, this calculator imposes a maximum limit on the "Number of Terms" you can request (e.g., 1000 terms). If you request more than the limit, it will display an error.

Q5: What happens if my formula results in an error for some 'n'?

The calculator evaluates the formula for each value of 'n' in the requested range. If an error occurs during evaluation (e.g., division by zero, logarithm of a negative number, syntax error in the formula itself), the calculation will stop, and an error message indicating the problem and the value of 'n' where it occurred will be displayed.

Q6: Can this calculator handle recursive sequences (like Fibonacci)?

No. This calculator works with explicit formulas where the n-th term (a_n) can be calculated directly from the index (n). It cannot handle recursive definitions where a term depends on previous terms (e.g., F_n = F_n-1 + F_n-2). Calculating recursive sequences requires a different approach, often involving loops that store previous values.