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Arithmetic Sequences & Series

In an arithmetic sequence the difference between consecutive terms is constant. That constant is called the common difference

d

d = T_n - T_{n-1}

Theorem 1📌 Learn for exam

General Term of an Arithmetic Sequence

T_n = a + (n-1)d

$a$ is the first term, $d$ is the common difference, $n$ is the position (term number).

▸ Arithmetic Sequence — drag a and dwatch the bars grow linearly

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Worked Example

Find $T_n$ and $T_{20}$ for the sequence $5, 9, 13, 17, \dots$

Worked Example

The $n$ th term of $3, 7, 11, 15, \dots$ is 195. Find $n$ .

Theorem 2📌 Learn for exam

Sum of an Arithmetic Series

S_n = \frac{n}{2}\big(2a + (n-1)d\big) \qquad \text{or} \qquad S_n = \frac{n}{2}(a + l)

Use the second form when the last term $l = T_n$ is already known — it saves a step.

Tip

Gauss's trick: pair the first and last term, the second and second-last, and so on — each pair adds to the same total,

a+l

. There are

n/2

such pairs.

Worked Example

Find the sum of the first 15 terms of $5, 9, 13, 17, \dots$