Everything Maths Grade 12

CHAPTER 3. SEQUENCES AND SERIES 3.5

• m is the lower bound (or start index), shown belowthe summation symbol;


• n is the upper bound (orend index), shown above the summation symbol;


• aiis a term of a sequence.

The index i increases from m to n in steps of 1.

If we are summing from i = 1 (which implies summing from the first term in asequence), then we can
use either Sn- or

�

-notation since they mean the same thing:

Sn=

�n

i=1

ai= a 1 +a 2 +··· +an (3.15)

For example, in the following sum,
�^5

i=1

i

we have to add togetherall the terms in the sequence ai= i from i = 1 up until i = 5:

�^5

i=1

i = 1 + 2 + 3 + 4 + 5 = 15

Examples

1.

�^6

i=1

2 i = 2^1 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6

= 2 + 4 + 8 + 16 + 32 + 64

= 126

2.

�^10

i=3

(3xi) = 3x^3 + 3x^4 +··· + 3x^9 + 3x^10

for any value x.

Some Basic Rules for Sigma Notation

1. Given two sequences, aiand bi,

�n

i=1

(ai+bi) =

�n

i=1

ai+

�n

i=1

bi (3.16)

2. For any constant c that is not dependent on the index i,

�n

i=1

c.ai = c.a 1 +c.a 2 +c.a 3 +··· +c.an

= c (a 1 +a 2 +a 3 +··· +an)

= c

�n

i=1

ai (3.17)