Everything Maths Grade 12

3.5 CHAPTER 3. SEQUENCES AND SERIES

the previous two terms.Hence, we can write down the recursive equation:

an= an− 1 +an− 2 (3.11)

3.5 Series

In this section we simply work on the concept of adding up the numbers belonging to arithmetic and

geometric sequences. We call the sum of any sequence of numbers a series.

Some Basics EMCV

If we add up the terms of a sequence, we obtainwhat is called a series. If we only sum a finitenumber

of terms, we get a finite series. We use the symbolSnto mean the sum of the firstn terms of a sequence

{a 1 ; a 2 ; a 3 ;... ;an}:

Sn= a 1 +a 2 +a 3 +··· +an (3.12)

For example, if we havethe following sequenceof numbers

1; 4; 9; 25; 36; 49; ...

and we wish to find thesum of the first four terms, then we write

S 4 = 1 + 4 + 9 + 25 = 39

The above is an example of a finite series sincewe are only summing four terms.

If we sum infinitely many terms of a sequence, we get an infinite series:

S∞= a 1 +a 2 +a 3 +... (3.13)

Sigma Notation EMCW

In this section we introduce a notation that willmake our lives a little easier.

A sum may be written out using the summationsymbol

�

. This symbol is sigma, which is the capital
letter “S” in the Greek alphabet. It indicates thatyou must sum the expression to the right of it:

�n

i=m

ai= am+am+1+··· +an− 1 +an (3.14)

where

• i is the index of the sum;