Series — the Sum Formula

Section 1 of 3

Sequence vs Series

A series is a sum

A sequence is a list: $2, 4, 6, 8, 10$. A series is that list added up: $2 + 4 + 6 + 8 + 10$. $S_n$ means the sum of the first $n$ terms.

Worked example — adding up terms

For $2 + 4 + 6 + 8 + 10 + \ldots$ find $S_3$ and $S_4$.

$S_3 = 2 + 4 + 6 = 12$

$S_4 = 2 + 4 + 6 + 8 = 20$

$S_3 = 12,\ S_4 = 20$

Section 2 of 3

The Sum Formula

From the Tables

$S_n = \dfrac{n}{2}\big\{\,2a + (n-1)d\,\big\}$. Fill in $a$, $d$ and $n$, then tidy up.

Worked example — find Sn and a sum

For $3 + 5 + 7 + 9 + \ldots$ find $S_n$ and hence $S_{20}$.

$a = 3$, $d = 2$

$S_n = \dfrac{n}{2}\{6 + 2(n-1)\} = \dfrac{n}{2}(2n + 4) = n(n+2)$

$S_{20} = 20(22) = 440$

$S_n = n(n+2),\ S_{20} = 440$

Worked example — a bigger sum

For $7 + 10 + 13 + \ldots$ find $S_n$ and hence $S_{70}$.

$a = 7$, $d = 3$

$S_n = \dfrac{n}{2}\{14 + 3(n-1)\} = \dfrac{n}{2}(3n + 11)$

$S_{70} = \dfrac{70}{2}(3(70) + 11) = 35(221) = 7735$

$S_{70} = 7735$

Section 3 of 3

Practice

You try

For $3 + 7 + 11 + 15 + \ldots$ find $S_n$.

$a = 3$, $d = 4$.

$S_n = \dfrac{n}{2}\{6 + 4(n-1)\} = \dfrac{n}{2}(4n + 2) = 2n^2 + n$

$S_n = 2n^2 + n$

You try

For $-6 - 2 + 2 + 6 + \ldots$ find $S_n$ and hence $S_{30}$.

$a = -6$, $d = 4$.

$S_n = \dfrac{n}{2}\{-12 + 4(n-1)\} = \dfrac{n}{2}(4n - 16) = 2n^2 - 8n$

$S_{30} = 2(900) - 8(30) = 1560$

$S_{30} = 1560$

You try

Find $S_{60}$ of $2 + 8 + 14 + \ldots$

$a = 2$, $d = 6$, $n = 60$.

$S_{60} = \dfrac{60}{2}\{2(2) + 6(59)\} = 30(358) = 10740$

$10740$

You try

Find $S_{33}$ of $-7 - 10 - 13 - \ldots$

$a = -7$, $d = -3$, $n = 33$.

$S_{33} = \dfrac{33}{2}\{2(-7) + 32(-3)\} = \dfrac{33}{2}(-110) = -1815$

$-1815$

You try

Find $S_{29}$ of $-15 - 11 - 7 - \ldots$

$a = -15$, $d = 4$, $n = 29$.

$S_{29} = \dfrac{29}{2}\{2(-15) + 28(4)\} = \dfrac{29}{2}(82) = 1189$

$1189$

That’s Class 6.

$S_n = \tfrac{n}{2}\{2a + (n-1)d\}$. Class 7: going backwards from a sum formula.