Sigma notation and arithmetic series

Warm-up

Three quick questions from earlier lessons.

Discover

To add $1 + 2 + \dots + 100$ , pair the first and last terms: $1 + 100$ , $2 + 99$ , … How many pairs, and what is each pair worth?

50 pairs of 101
100 pairs of 101

Key idea

$\displaystyle\sum_{k=p}^{q} a_k$ means $a_p + a_{p+1} + \dots + a_q$ : it has $q - p + 1$ terms.

The sum of the

n

terms of an arithmetic sequence is

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

: the number of terms times the average of the first and last. In particular

1 + 2 + \dots + n = \dfrac{n(n + 1)}{2}

.

Worked example

Find $\displaystyle\sum_{k=1}^{20} (3k + 2)$ .

First term

5

, last term

62

k = 1 and k = 20.

S = \dfrac{20(5 + 62)}{2}

20 terms.

S = 670

10 · 67.

Your turn, guided

Your turn: $2 + 4 + 6 + \dots + 40$ .

n = 20

40 ÷ 2: 20 terms.

S = 420

20 · 42 / 2.

Practice ★

How many terms does $\displaystyle\sum_{k=3}^{ 17 } (2k + 1)$ have?
How many terms does $\displaystyle\sum_{k=3}^{ 21 } (2k + 1)$ have?
How many terms does $\displaystyle\sum_{k=2}^{ 38 } (2k + 1)$ have?

Watch out!

Practice ★★

Find $\displaystyle\sum_{k=1}^{ 30 } (3k + 7)$ .
Find $\displaystyle\sum_{k=1}^{ 11 } (6k + 8)$ .
Find $\displaystyle\sum_{k=1}^{ 21 } (6k + 6)$ .

Find $1 + 2 + 3 + \dots + 94$ .
Find $1 + 2 + 3 + \dots + 80$ .

Exit ticket

Three questions to check you have got it.

Find $\displaystyle\sum_{k=1}^{ 30 } (3k + 7)$ .
Find $\displaystyle\sum_{k=1}^{ 11 } (6k + 8)$ .
Find $\displaystyle\sum_{k=1}^{ 21 } (6k + 6)$ .

Summary card

Arithmetic series

$\displaystyle\sum_{k=p}^{q}$ has $q - p + 1$ terms.
$S_n = \dfrac{n(a_1 + a_n)}{2}$ .
$1 + 2 + \dots + n = \dfrac{n(n + 1)}{2}$ .

Watch out: count the terms with $+1$ .

About this lesson