3n

3n is an algebraic expression representing three times a quantity n. When n is an integer, 3n denotes a member of the set of multiples of 3. In particular, the set {3n : n ∈ Z} comprises all integers divisible by 3, such as ..., -6, -3, 0, 3, 6, 9, ...

Key properties follow from this definition. 3n is always divisible by 3. If n is any integer, 3n has the same parity as n (odd n yields an odd 3n, even n yields an even 3n). As a function, f(n) = 3n is linear with slope 3 and intercept 0, so it grows or decreases at three times the rate of n.

In modular arithmetic, 3n is congruent to 0 modulo 3 for all integers n, reflecting its divisibility by 3. More broadly, 3n can be viewed as an arithmetic progression with common difference 3.

Examples illustrate the concept: when n = 0, 3n = 0; n = 1 gives 3; n = 2 gives 6; and n = -4 gives -12. If n is restricted to natural numbers, 3n yields nonnegative multiples of 3.

Generalizations include treating 3n as a special case of a linear function or as a scaling by a constant factor. In contexts where n is not restricted to integers, 3n remains a well-defined real number, though the interpretation as a multiple of 3 applies specifically to integer n.