Arithmetic Sequences and Series

Sequences and seriesArithmetic Sequences and SeriesGeometric Sequences and Series Special Sums and Means

Terms

Sequence

a, a + d, a + 2 d, a + 3 d, \dots

d common difference

Common difference

d = a_{2} - a_{1} = \frac{a_{n} - a_{m}}{n - m}

nth term

a_{n} = a_{1} + (n - 1) d = a_{m} + (n - m) d

kth term from the end

l - (k - 1) d

l the last term

Recursive rule

a_{n} = a_{n - 1} + d

n ≥ 2

Number of terms

n = \frac{l - a_{1}}{d} + 1

Sum of n terms

S_{n} = \frac{n}{2} (a_{1} + a_{n})

Sum of n terms, from d

S_{n} = \frac{n}{2} (2 a_{1} + (n - 1) d)

Term from sums

a_{n} = S_{n} - S_{n - 1}

n ≥ 2

Mean of two numbers

\frac{a + b}{2}

Three terms in a row

a, b, c arithmetic \Leftrightarrow 2 b = a + c

k means between a and b

d = \frac{b - a}{k + 1}

Sum of the k means

k \cdot \frac{a + b}{2}

Three terms, centered

a - d, a, a + d

Four terms, centered

a - 3 d, a - d, a + d, a + 3 d

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