Explicit Formula Calculator

Explicit Formula:

\[ a_n = a_1 + (n - 1) \times d \quad \text{(Arithmetic)} \] \[ a_n = a_1 \times r^{(n-1)} \quad \text{(Geometric)} \]

Sequence Type:

First Term (a₁):

unit of sequence

Common Difference (d):

unit of sequence

Common Ratio (r):

unitless

Term Number (n):

unitless

nth Term (aₙ):

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1. What is the Explicit Formula?

The explicit formula allows direct calculation of any term in a sequence without needing to know previous terms. For arithmetic sequences, it uses the first term and common difference. For geometric sequences, it uses the first term and common ratio.

2. How Does the Calculator Work?

The calculator uses the explicit formulas:

\[ a_n = a_1 + (n - 1) \times d \quad \text{(Arithmetic Sequence)} \] \[ a_n = a_1 \times r^{(n-1)} \quad \text{(Geometric Sequence)} \]

Where:

\( a_n \) — nth term of the sequence
\( a_1 \) — first term of the sequence
\( n \) — term number
\( d \) — common difference (arithmetic sequences)
\( r \) — common ratio (geometric sequences)

Explanation: The formulas provide a direct way to calculate any term in the sequence based on its position and the sequence's defining characteristics.

3. Importance of Explicit Formulas

Details: Explicit formulas are essential for quickly finding terms in sequences, analyzing sequence patterns, and solving problems in mathematics, computer science, and various scientific fields.

4. Using the Calculator

Tips: Select the sequence type, enter the first term, common difference/ratio, and the term number you want to calculate. All values must be valid numbers.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between arithmetic and geometric sequences?
A: Arithmetic sequences have a constant difference between terms, while geometric sequences have a constant ratio between terms.

Q2: Can I use this for negative term numbers?
A: No, term numbers must be positive integers starting from 1.

Q3: What if the common ratio is negative?
A: The calculator handles negative common ratios, producing alternating positive and negative terms.

Q4: Are there limitations to these formulas?
A: These formulas work for standard arithmetic and geometric sequences but not for more complex recursive sequences.

Q5: Can I calculate multiple terms at once?
A: This calculator calculates one specific term at a time. For multiple terms, you would need to calculate each separately.