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Product Notation / Capital Pi — Definition, Formula & Examples

Product notation uses the capital Greek letter pi (∏) to write the product of a sequence of terms in a compact way, just as sigma notation (Σ) represents a sum. Each factor in the product is generated by substituting consecutive index values into a given expression.

For an integer-valued index ii running from mm to nn where m≤nm \le n, the product ∏i=mnai\displaystyle\prod_{i=m}^{n} a_i is defined as am⋅am+1⋅am+2⋯ana_m \cdot a_{m+1} \cdot a_{m+2} \cdots a_n. When m>nm > n, the empty product is defined to equal 11.

Key Formula

∏i=mnai=am⋅am+1⋅am+2⋯an\prod_{i=m}^{n} a_i = a_m \cdot a_{m+1} \cdot a_{m+2} \cdots a_n
Where:
  • ii = Index variable that takes consecutive integer values
  • mm = Lower bound (starting value of the index)
  • nn = Upper bound (ending value of the index)
  • aia_i = The expression evaluated at each index value

How It Works

Read the notation bottom-to-top: the variable and its starting value appear below the ∏ symbol, and the upper bound sits above it. You evaluate the expression for every integer value of the index from the lower bound to the upper bound, then multiply all of those results together. If the lower bound exceeds the upper bound, the result is the empty product, which equals 11 by convention.

Worked Example

Problem: Evaluate ∏k=15(2k)\displaystyle\prod_{k=1}^{5} (2k).
List the factors: Substitute k = 1, 2, 3, 4, 5 into the expression 2k.
2(1),  2(2),  2(3),  2(4),  2(5)=2,  4,  6,  8,  102(1),\; 2(2),\; 2(3),\; 2(4),\; 2(5) = 2,\; 4,\; 6,\; 8,\; 10
Multiply all factors: Compute the product of these five values.
2×4×6×8×10=3,8402 \times 4 \times 6 \times 8 \times 10 = 3{,}840
Answer: ∏k=15(2k)=3,840\displaystyle\prod_{k=1}^{5}(2k) = 3{,}840

Why It Matters

Product notation appears throughout combinatorics (factorials, binomial coefficients), linear algebra (determinants), and probability (likelihoods of independent events). Writing these multiplications compactly with ∏ is essential for reading and writing proofs in any course beyond introductory calculus.

Common Mistakes

Mistake: Confusing ∏ (product) with Σ (sum) and adding the terms instead of multiplying them.
Correction: Remember that capital pi (∏) always means multiply. Capital sigma (Σ) means add. A quick mnemonic: Pi → Product, Sigma → Sum.

Related Terms