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Arithmetic sequence and series

In an arithmetic sequence the difference between neighbouring terms is constant. Term number n is the first term plus the difference added n minus 1 times. The sum of the first n terms is the number of terms times the average of the first and the last. You therefore find the last term first. A constant difference, not a constant factor, is the sign to look for.

Try it yourself

Geometric series

a_n = a_1 + (n-1)d

term number n

S_n = \frac{n(a_1 + a_n)}{2}

the sum

Symbols

$a_1$	the first term
$d$	the difference
$n$	the number of terms

Example

$a_1 = 3$ and $d = 4$ give $a_{10} = 3 + 9 \cdot 4 = 39$ .

The sum is $\dfrac{10(3 + 39)}{2} = 210$ .

There are n minus 1 steps from the first term to term number n.

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Part of Mathematics S2: Sequences and series.