Resources · Mathematics
Explicit Formula
Explicit formulas represent the most direct and efficient method to analyze numeric sequences. Unlike step-by-step methods, explicit rules find the exact value at any point in a pattern. For instant progression generation, visit our arithmetic sequence calculator or check out our guide on the recursive formula.
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What is an explicit formula?
An explicit formula is a direct algebraic relationship that expresses the value of a term an based solely on its position n. Think of it like a function where the input is the term number (like index 1, 2, or 100) and the output is the value at that position.
The two most common types of explicit formulas describe arithmetic (constant addition) and geometric (constant multiplication) patterns:
Standard Explicit Equations
| Sequence Type | Explicit Equation | Variables Explained |
|---|---|---|
| Arithmetic | an = a1 + (n − 1)d | a₁ = first term, n = position, d = common difference |
| Geometric | an = a1 × rn−1 | a₁ = first term, n = position, r = common ratio |
Worked Examples
Example 1: Finding a High-Index Term in an Arithmetic Sequence
For the sequence starting at 5 with a common difference of 3, find the 50th term.
- Formula: an = a1 + (n − 1)d
- Given: a₁ = 5, d = 3, n = 50
- Calculation: a₅₀ = 5 + (50 − 1) × 3 = 5 + 49 × 3 = 5 + 147
- Result: a₅₀ = 152
Example 2: Geometric Sequence Calculations
For a geometric sequence starting at 2 with a common ratio of 3, find the 5th term.
- Formula: an = a1 × rn−1
- Given: a₁ = 2, r = 3, n = 5
- Calculation: a₅ = 2 × 35−1 = 2 × 3⁴ = 2 × 81
- Result: a₅ = 162
Common Mistakes with Explicit Formulas
- Order of Operations Violations (Geometric): When evaluating a1 × rn−1, you must calculate the exponent rn−1 first before multiplying by the starting term a1. For example, 2 × 3⁴ is 2 × 81 = 162, not (2 × 3)⁴ = 6⁴ = 1,296.
- Using n instead of n-1: Remember that the first term a1 has not undergone any steps of addition or multiplication yet. Therefore, the formula adjusts the position by using n-1. Using n instead will shift the entire sequence forward by one index.
- Mixing up index and value: Remember that n represents the integer index (like the 10th step), whereas an represents the numerical value at that index.
Run the numbers
Convert sequence types or perform series calculations using our resources: