8n

8n is a mathematical notation for the product of eight and a variable n, where n is typically an integer or natural number. As a function of n, f(n) = 8n is a linear function with slope 8 and zero intercept; its graph is a straight line through the origin. The expression is commonly used to describe sequences, sets, or formulas that scale linearly with n.

In number theory, the set {8n : n ∈ Z} consists of all integers divisible by 8, i.e., the multiples of 8. If n ranges over natural numbers, the positive multiples of 8 are 8, 16, 24, and so on. This makes 8n a representative of an arithmetic progression with first term 0 and common difference 8. Consequently, 8n is always even and, for any integer n, satisfies 8n ≡ 0 (mod 8). The identity 8(n + m) = 8n + 8m reflects the distributive nature of multiplication over addition.

Applications and related concepts include its interpretation as a linear growth model, a basic example of a homogeneous linear term, and its use in modular arithmetic and divisibility discussions. In computing, 8n can be computed efficiently as a left shift by three bits: 8n = n << 3, illustrating a link between arithmetic expressions and binary operations.

Examples: n = 0 gives 0, n = 1 gives 8, n = -3 gives -24. The term 8n frequently appears in algebraic contexts as a simple, scalable, integer-valued expression.