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Equations for Sequences
Arithmetic Sequences
The (explicit) equation for an arithmetic sequence is: t(n) = mn + b where n is the term number, m is the common difference, and b is the zeroth term (also known as the starting point or initial value). These equations are similar to a continuous linear function f(x) = mx + b, where m is the slope (growth) and b is the y‑intercept (starting point).
For example, the arithmetic sequence 10, 13, 16, 19, … can be represented by t(n) = 3n + 7. (Note that 10 is the first term of this sequence, and 7 is the zeroth term.)
An alternative notation for arithmetic sequences is an = mn + a0 where n is the term number, m is the common difference, and a0 is the zeroth term. Using this alternative notation, the equation for the sequence 10, 13, 16, 19, … is written an = 3n + 7.
Geometric Sequences
The (explicit) equation for a geometric sequence is: t(n) = abn, where n is the term number, b is the multiplier or common ratio (sequence generator), and a is the zeroth term. An alternative notation for geometric sequences is an = a0 · bn where n is the term number, b is the common ratio, and a0 is the zeroth term.
For example, the geometric sequence 6, 18, 54, … can be represented by t(n) = 2 · 3n or by an = 2 · 3n.
Recursive Sequences
A recursive sequence is a sequence in which each term depends on the term(s) before it. The equation of a recursive sequence requires at least one term to be specified. A recursive sequence can be arithmetic, geometric, or neither.
For example, the sequence 3, 11, 123, 15131, … can be defined by the recursive equation:
t(1) = 3, t(n + 1) = (t(n))2 + 2
An alternative notation for the equation of the sequence above is:
a1 = 3, an+1 = (an)2 + 2
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