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πŸ”’ Number Sequence Calculator

Analyze arithmetic, geometric, and simple pattern sequences

Arithmetic Sequence

Nth Term:

Geometric Sequence

Nth Term:

Pattern Detection

Sequence Analysis Framework

Lean reference to classify, extend, and sanity‑check numeric sequences.

1. Core Families

Arithmetic: a_n = a_1 + (nβˆ’1)d
Geometric: a_n = a_1 r^{nβˆ’1}
Quadratic: a_n = AnΒ²+Bn+C (2nd diff = 2A)
Fibonacci-type: a_n = a_{nβˆ’1}+a_{nβˆ’2}
Exponential: a_n = A b^{n} + C

2. Rapid Classification

  1. First differences constant β†’ arithmetic.
  2. Ratios constant (no zeros) β†’ geometric.
  3. Second differences constant β†’ quadratic.
  4. Check simple recurrences (sum of previous, linear combo).
  5. Look for sign flip (multiply by (βˆ’1)^n to simplify).

3. Heuristics

  • Explosive early growth usually geometric/exponential.
  • Slow curved growth with stable 2nd diff β†’ polynomial.
  • Single anomaly? Possibly data error not new rule.

4. Guardrails

  • Re-generate known terms using inferred formula.
  • Compare exact vs rounded ratios to avoid misclassifying.
  • Estimate size of a large n term β†’ sanity (overflow?).

5. Pitfalls

  • Overfitting high-degree polynomial to few points.
  • Index off-by-one (n vs position starting at 0).
  • Assuming ratio with zeros / negative alternation.

6. Quick Reference

S_arith = n/2 (2a₁ + (nβˆ’1)d) | S_geom = a₁(1βˆ’r^n)/(1βˆ’r)
Triangular T_n = n(n+1)/2 | Factorial n! | F_n β‰ˆ Ο†^n/√5

7. FAQ

  • Sequence vs series? Series sums sequence terms.
  • Infinite sum exists? For |r|<1 geometric; others need tests.
  • Multiple fits? Prefer simplest consistent rule.

8. Action Tip

Write the rule immediately (e.g., a_n = 7 + 5(nβˆ’1)) to avoid reinterpretation later.

}); } catch (error) { updateResults('pattern', { error: 'Error analyzing pattern. Please check your input format.' }); } } function updateResults(type, data) { const sequenceDisplay = document.getElementById('sequenceDisplay'); const conversionResults = document.getElementById('conversionResults'); const solutionSteps = document.getElementById('solutionSteps'); if (data.error) { sequenceDisplay.innerHTML = `${data.error}`; return; } if (type === 'arithmetic') { sequenceDisplay.innerHTML = ` Arithmetic Sequence:
${data.sequence.join(', ')}${data.sequence.length < data.numTerms ? '...' : ''}

Formula: ${data.formula} `; conversionResults.innerHTML = `
${data.nthPosition}th Term:
${data.nthTerm.toFixed(2)}
Sum of ${data.numTerms} Terms:
${data.sum.toFixed(2)}
First Term (a₁):
${data.a1}
Common Difference (d):
${data.d}
Last Term (a${data.numTerms}):
${data.sequence[data.sequence.length - 1].toFixed(2)}
Average Term:
${(data.sum / data.numTerms).toFixed(2)}
`; solutionSteps.innerHTML = `
  1. Given: First term a₁ = ${data.a1}, Common difference d = ${data.d}
  2. Formula: aβ‚™ = a₁ + (n-1)d
  3. Calculate ${data.nthPosition}th term: a${data.nthPosition} = ${data.a1} + (${data.nthPosition}-1) Γ— ${data.d}
  4. Simplify: a${data.nthPosition} = ${data.a1} + ${(data.nthPosition-1) * data.d} = ${data.nthTerm.toFixed(2)}
  5. Sum formula: Sβ‚™ = n/2 Γ— (2a₁ + (n-1)d)
  6. Sum of ${data.numTerms} terms: S${data.numTerms} = ${data.numTerms}/2 Γ— (2Γ—${data.a1} + ${data.numTerms-1}Γ—${data.d}) = ${data.sum.toFixed(2)}
`; } else if (type === 'geometric') { sequenceDisplay.innerHTML = ` Geometric Sequence:
${data.sequence.map(x => x.toLocaleString()).join(', ')}${data.sequence.length < data.numTerms ? '...' : ''}

Formula: ${data.formula} `; conversionResults.innerHTML = `
${data.nthPosition}th Term:
${data.nthTerm.toLocaleString()}
Sum of ${data.numTerms} Terms:
${data.sum.toLocaleString()}
First Term (a₁):
${data.a1}
Common Ratio (r):
${data.r}
Growth Factor:
${Math.pow(data.r, data.numTerms - 1).toLocaleString()}
Convergence:
${Math.abs(data.r) < 1 ? 'Converges' : Math.abs(data.r) > 1 ? 'Diverges' : 'Constant'}
`; solutionSteps.innerHTML = `
  1. Given: First term a₁ = ${data.a1}, Common ratio r = ${data.r}
  2. Formula: aβ‚™ = a₁ Γ— r^(n-1)
  3. Calculate ${data.nthPosition}th term: a${data.nthPosition} = ${data.a1} Γ— ${data.r}^(${data.nthPosition}-1)
  4. Simplify: a${data.nthPosition} = ${data.a1} Γ— ${data.r}^${data.nthPosition-1} = ${data.a1} Γ— ${Math.pow(data.r, data.nthPosition-1).toLocaleString()} = ${data.nthTerm.toLocaleString()}
  5. Sum formula: Sβ‚™ = a₁(rⁿ - 1)/(r - 1) for r β‰  1
  6. Sum of ${data.numTerms} terms: S${data.numTerms} = ${data.a1}(${data.r}^${data.numTerms} - 1)/(${data.r} - 1) = ${data.sum.toLocaleString()}
`; } else if (type === 'pattern') { sequenceDisplay.innerHTML = ` Pattern Analysis:
Sequence: ${data.sequence.join(', ')}
Pattern Type: ${data.patternType}
Next Term: ${data.nextTerm}
Formula: ${data.formula} `; let analysisHTML = `
Pattern Type:
${data.patternType}
Next Term:
${data.nextTerm}
First Differences:
${data.differences.map(d => d.toFixed(2)).join(', ')}
`; if (data.ratios.length > 0) { analysisHTML += `
Ratios:
${data.ratios.map(r => r.toFixed(4)).join(', ')}
`; } if (data.secondDifferences.length > 0) { analysisHTML += `
Second Differences:
${data.secondDifferences.map(d => d.toFixed(2)).join(', ')}
`; } analysisHTML += `
Properties:
${data.isArithmetic ? 'βœ“ Arithmetic' : 'βœ— Not Arithmetic'}
${data.isGeometric ? 'βœ“ Geometric' : 'βœ— Not Geometric'}
${data.isQuadratic ? 'βœ“ Quadratic' : 'βœ— Not Quadratic'}
`; conversionResults.innerHTML = analysisHTML; let stepsHTML = '
    '; stepsHTML += '
  1. Input Sequence: ' + data.sequence.join(', ') + '
    2. '; stepsHTML += '
  2. Calculate First Differences: ' + data.differences.map(d => d.toFixed(2)).join(', ') + '
    3. '; if (data.isArithmetic) { stepsHTML += '
  3. Constant Differences Found: This is an arithmetic sequence
    4. '; stepsHTML += '
  4. Common Difference: d = ' + data.differences[0] + '
    5. '; } else { stepsHTML += '
  5. Check Ratios: ' + data.ratios.map(r => r.toFixed(4)).join(', ') + '
    6. '; if (data.isGeometric) { stepsHTML += '
  6. Constant Ratios Found: This is a geometric sequence
    7. '; stepsHTML += '
  7. Common Ratio: r = ' + data.ratios[0] + '
    8. '; } else if (data.isQuadratic) { stepsHTML += '
  8. Check Second Differences: ' + data.secondDifferences.map(d => d.toFixed(2)).join(', ') + '
    9. '; stepsHTML += '
  9. Constant Second Differences Found: This is a quadratic sequence
    10. '; } else { stepsHTML += '
  10. No Standard Pattern Found: May be a more complex sequence
    11. '; } } stepsHTML += '
  11. Next Term Prediction: ' + data.nextTerm + '
    12. '; stepsHTML += '
'; solutionSteps.innerHTML = stepsHTML; } } // Initialize calculations calculateArithmetic(); calculateGeometric(); analyzePattern();