Product of the First 20 Odd Numbers, Exact Value

Capital pi notation, evaluated term by term

The symbol ∏ asks you to multiply every term of a sequence from index m to index n. This product notation calculator does that for an arithmetic sequence, a geometric sequence or any formula in n, and keeps the answer exact: whole numbers to the last digit, fractions as fractions. Four results it gives, each checked against the calculator:

11/20

∏ (1 − 1/n²) for n = 2 to 10

26 digits

in 1 × 2 × ... × 25, printed in full

0.492703

chance that 23 people all have different birthdays

512

terms 3 to 5 of 1, 2, 4, 8, ...: 4 × 8 × 16

Three choices and two indexes

Pick the kind of sequence. Arithmetic adds the same d each step (1, 3, 5, 7), geometric multiplies by the same r (3, 6, 12), and a formula covers everything else.
Describe the terms. For arithmetic and geometric sequences type the first term a₁ and d or r; fractions such as 1/2 are fine. For a formula write it in n, for example 1 - 1/n^2, 2n + 1 or (366 - n)/365.
Set the range. The first index m and the last index n are whole numbers, up to 1000 terms. Arithmetic and geometric terms are counted from 1; a formula may start at 0 or below.
Read the result: the product, the sign with the number of negative terms, the size in digits, a closed form for the two standard sequences, and a table with the running product after each term.

How a product of terms behaves

Capital pi is the multiplying twin of capital sigma. Where Σ from i = 1 to 5 of i gives 15, ∏ from i = 1 to 5 of i gives 120, which is 5!. That one example shows the two ways products differ from sums. They grow far faster, so 25 small whole numbers already make a 26-digit number, and a single zero anywhere in the range wipes the whole product out.

The sign follows a simple count: an odd number of negative terms gives a negative product, an even number a positive one. The terms −5, −3, −1, 1, 3, 5 contain three negatives, so their product is −225. Fractions pull the other way: a product of numbers just below 1 shrinks slowly, and many products of the form 1 ± something telescope, meaning each numerator cancels the next denominator. That is why ∏ (1 + 1/n) from 1 to 99 comes out as exactly 100.

For a geometric sequence there is always a closed form. Multiplying terms m to n of a₁rⁱ⁻¹ gives a₁^k times r raised to the sum of the exponents, where k is the number of terms. For 3, 3/2, 3/4, ... the first six terms give 3⁶ · (1/2)¹⁵ = 729/32768. An arithmetic sequence has no such tidy formula in general, but factoring d out of every term turns it into a rising run of numbers, which becomes a ratio of factorials whenever a_m/d is a whole number.

Products with a closed form

Product	Closed form	Value at n = 10
∏ i, i = 1 to n	n!	3,628,800
∏ 2i (even numbers)	2ⁿ n!	3,715,891,200
∏ (2i − 1) (odd numbers)	(2n)! / (2ⁿ n!)	654,729,075
∏ (1 + 1/i)	n + 1	11
∏ (1 − 1/i²), i = 2 to n	(n + 1) / (2n)	11/20
∏ a₁rⁱ⁻¹	a₁ⁿ r^n(n−1)/2	2⁴⁵ for a₁ = 1, r = 2
∏ 4i² / (4i² − 1)	tends to π/2 (Wallis)	1.53385

Facts that make products unlike sums

The birthday product. The chance that n people share no birthday is ∏ (366 − i)/365 for i = 1 to n. It is 0.524305 for 22 people and drops to 0.492703 for 23, the first group size where a shared birthday is more likely than not.

Telescoping hides in plain sight. ∏ (1 − 1/n²) from 2 to 100 looks like 99 messy fractions, yet it equals 101/200. Each factor splits into (n − 1)/n times (n + 1)/n, and almost everything cancels.

Wallis creeps toward π/2. Ten factors of 4n²/(4n² − 1) give 1.53385, a thousand give 1.5704, and π/2 is 1.5708. An infinite product can converge, but slowly.

An empty product equals 1. When the upper index is below the lower one there is nothing to multiply, and by convention the result is 1, the same way 0! = 1. The calculator flags such a range rather than printing 1 silently, since a swapped m and n is usually a typo.

Logarithms turn ∏ into Σ. log(∏ a_i) = Σ log a_i. The log₁₀ tile uses this: 25! has log₁₀ 25.190646, so it has 26 digits.

Who multiplies a sequence

Students of discrete math and precalculus checking a ∏ exercise, a telescoping product or a proof by induction against an exact value.

Probability learners computing "no two alike" chances, such as the birthday problem or drawing without replacement, which are products of shrinking fractions.

Combinatorics work with double factorials and falling factorials: 1 × 3 × 5 × ... × 19 is an arithmetic product with d = 2.

Programmers testing a loop that multiplies, who need the exact value to compare against rather than a rounded one that overflowed.

Five products run through the calculator

What you type	Range	Result
Arithmetic, a₁ = 1, d = 1	1 to 25	15,511,210,043,330,985,984,000,000
Arithmetic, a₁ = −5, d = 2	1 to 6	−225 (three negative terms)
Geometric, a₁ = 3, r = 1/2	1 to 6	729/32768 ≈ 0.0222473
Formula 1 + 1/n	1 to 99	100
Formula (366 − n)/365	1 to 23	0.492703

Pi notation questions

What can I type in the formula box?

Numbers, the index n (i and k work too), + − * / and ^ for powers, brackets, an exclamation mark for a factorial (n!), sqrt() and abs(). A number next to n or a bracket multiplies it, so 2n + 1 and n(n + 1) both work, and −n^2 means −(n²). A square root that is not a perfect square switches the answer to a decimal.

What happens if a term divides by zero?

The calculator names the index where it happens. For 1/(n − 3) from 1 to 5 it stops at n = 3, since that term does not exist; start the range at 4 or change the formula.

Is there a formula for the product of an arithmetic sequence?

Not a simple one in general. Write each term as d(x + j) with x = a_m/d; the product is d^k times x(x + 1)...(x + k − 1), a rising factorial that the Gamma function extends to any x. When x is a whole number it becomes a ratio of ordinary factorials, and the result box writes that ratio out.

How is this different from multiplying a list of numbers?

A list tool multiplies numbers you paste in. Here you describe the rule that generates the terms and the index range, so a thousand terms take one line, and the closed forms and the running product show how the answer is built.

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What is the product of the first 20 odd numbers?

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