ALGEBRAArithmeticMathematics Calculator
Σ

Summation: Σ f(i) from start to end

Sigma notation Σ sums f(i) for i from start to end. Σi = n(n+1)/2, Σi² = n(n+1)(2n+1)/6, Σi³ = [n(n+1)/2]². Use i as variable in expression.

Concept Fundamentals
n(n+1)/2
Σi
n(n+1)(2n+1)/6
Σi²
[n(n+1)/2]²
Σi³
Variable
i

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Σi from 1 to n = n(n+1)/2. Triangular numbers. Σi² = n(n+1)(2n+1)/6. Σi³ = [n(n+1)/2]². Use i in expression: e.g. 2*i for 2,4,6,...

Key quantities
n(n+1)/2
Σi
Key relation
n(n+1)(2n+1)/6
Σi²
Key relation
[n(n+1)/2]²
Σi³
Key relation
Variable
i
Key relation

Ready to run the numbers?

Why: Summation notation compactly expresses series. Σi = 1+2+...+n. Σi², Σi³ have closed forms. Used in calculus, sequences, and discrete math.

How: Evaluate f(i) for each i from start to end, sum. Use i as variable. Common: Σi = n(n+1)/2, Σi² = n(n+1)(2n+1)/6, Σi³ = [n(n+1)/2]².

Σi from 1 to n = n(n+1)/2. Triangular numbers.Σi² = n(n+1)(2n+1)/6. Σi³ = [n(n+1)/2]².
Sources:Khan AcademyNCTM Standards

Run the calculator when you are ready.

Calculate SummationEnter expression and range
sigma.sh
CALCULATED
$ sigma --expr=i --start=1 --end=10
Sum
55
Terms
10
Formula
Σi = n(n+1)/2
Range
1 to 10
Summation Calculator
Σ(i) = 55
Σi = n(n+1)/2
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Terms (first 20)

📐 Step-by-Step Breakdown

SETUP
Expression
Σ(i)
METHOD
Range
i = 1 to 10
RESULT
Sum
55
Formula match
Σi = n(n+1)/2
Terms
1, 2, 3, 4, 5, 6, 7, 8, 9, 10

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

Σ

Σi = n(n+1)/2

— Sum 1 to n

📐

Σi² = n(n+1)(2n+1)/6

— Sum of squares

📋 Key Takeaways

  • • Sigma notation: Σ f(i) from i=start to end = f(start)+f(start+1)+...+f(end)
  • • Σi = n(n+1)/2 — Gauss formula for sum of 1 to n
  • • Σi² = n(n+1)(2n+1)/6 — sum of squares
  • • Σi³ = [n(n+1)/2]² — sum of cubes
  • • Use i as index variable. Support: i, i^2, 2*i+1, etc.

💡 Did You Know?

∑Σ (sigma) is the Greek letter S, standing for sum. Euler popularized it.Source: Mathematical Notation
🧒Σi from 1 to n = 1+2+...+n = n(n+1)/2 — Gauss discovered this as a child.Source: Mathematical History
📐Σi² from 1 to n = n(n+1)(2n+1)/6 — used in variance and integration.Source: Statistics
📊Arithmetic series: Σ(a+(i−1)d) = n/2[2a+(n−1)d].Source: Series Formulas
📈Geometric: Σar^(i−1) = a(1−r^n)/(1−r) for r≠1.Source: Geometric Series
∫Calculus: Riemann sums use Σ to approximate definite integrals.Source: Analysis

📖 How It Works

Substitute i = start, start+1, ..., end into the expression. Add all terms. For common patterns (Σi, Σi², Σi³, Σ1) starting at 1, we compare with closed-form formulas.

Σi from 1 to 5 = 1+2+3+4+5 = 15 = 5(6)/2

📝 Worked Example: Σi² from 1 to 5

Terms: 1² + 2² + 3² + 4² + 5² = 1 + 4 + 9 + 16 + 25 = 55

Formula: n(n+1)(2n+1)/6 = 5×6×11/6 = 55 ✓

🚀 Real-World Applications

📊 Variance

Σ(xᵢ - μ)² for standard deviation.

📐 Riemann Sums

Approximate ∫f(x)dx with Σf(xᵢ)Δx.

💰 Finance

Sum of payments, NPV calculations.

🔬 Physics

Sum of forces, moments, energy.

📈 Statistics

Sum of squared errors, MSE.

🎯 Algorithms

Time complexity: Σ loop iterations.

⚠️ Common Mistakes to Avoid

  • Wrong variable: Use i (lowercase) as the index. The calculator substitutes i with each value.
  • i^2 vs i**2: Both work. Use * for multiplication: 2*i+1.
  • Start at 0: Formulas like n(n+1)/2 assume start=1. For start=0, Σi = n(n-1)/2.
  • Large range: Max 1000 terms to avoid performance issues.
  • Invalid expression: Only use safe math: i, +, -, *, /, ^, **, parentheses.

🎯 Expert Tips

💡 Use Formulas

When available, formulas are faster and exact. Compare with computed sum.

💡 Power Notation

Use i^2 or i**2 for squares. The calculator converts ^ to **.

💡 Index Start

Index can start at 0 or 1. Adjust formula accordingly for closed forms.

💡 Arithmetic Series

Use a+(i-1)*d. Formula: n/2[2a+(n-1)d].

📊 Reference Table

SeriesFormula
Σi (1 to n)n(n+1)/2
Σi² (1 to n)n(n+1)(2n+1)/6
Σi³ (1 to n)[n(n+1)/2]²
Σ1 (1 to n)n

📐 Quick Reference

Σ
Sigma (sum)
i
Index variable
55
Σi 1 to 10
5050
Σi 1 to 100

🎓 Practice Problems

Σi from 1 to 20 → Answer: 210
Σi² from 1 to 5 → Answer: 55
Σ(2i+1) from 0 to 4 → Answer: 25
Σi³ from 1 to 4 → Answer: 100

❓ FAQ

What is sigma notation?

Σ denotes sum. Σf(i) from i=m to n = f(m)+f(m+1)+...+f(n).

How to enter i²?

Use i^2 or i**2. The calculator substitutes i with each index value.

Can start be 0?

Yes. Formulas may differ (e.g. Σi from 0 to n = n(n+1)/2 with n+1 terms).

Max range?

1000 terms to avoid performance issues.

Arithmetic series?

Use expression like a+(i-1)*d. Formula: n/2[2a+(n-1)d].

What is Σi from 1 to 100?

5050. Using Gauss formula: 100×101/2 = 5050.

Why use formulas?

Faster, exact, and verifies your computed sum. We auto-detect Σi, Σi², Σi³, Σ1.

📌 Summary

Sigma notation Σf(i) sums f(i) over i from start to end. Common formulas: Σi = n(n+1)/2, Σi² = n(n+1)(2n+1)/6, Σi³ = [n(n+1)/2]². Use i as the index variable. The calculator compares with formulas when applicable.

✅ Verification Tip

For Σi, Σi², Σi³, Σ1 from 1 to n, we show formula match. If it matches, your sum is correct. Manually add a few terms for other expressions.

🔗 Next Steps

Explore the Arithmetic Sequence Calculator for linear series, or the Geometric Sequence Calculator for multiplicative growth. The Average Calculator uses summation.

⚠️ Disclaimer: Expression is evaluated with Function(). Use only safe math (i, +, -, *, /, ^, **). Educational use.

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