Everything Maths Grade 12

3.6 CHAPTER 3. SEQUENCES AND SERIES

Exercise 3 - 2

1. What is

�^4

k=1

2?

2. Determine

�^3

i=− 1

i.

3. Expand

�^5

k=0

i.

4. Calculate the value of a if:
   �^3

k=1

a. 2 k−^1 = 28

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(1.) 01cf (2.) 01cg (3.) 01ch (4.) 01ci

3.6 Finite Arithmetic Series

Remember that an arithmetic sequence is a sequence of numbers, suchthat the difference between

any term and the previous term is a constant number, d, called the constant difference:

an= a 1 +d (n− 1) (3.18)

where

• n is the index of the sequence;


• anis the nth-term of the sequence;


• a 1 is the first term;


• d is the common difference.

When we sum a finite number of terms in an arithmetic sequence, we get a finite arithmetic series.

A simple arithmetic sequence is when a 1 = 1 and d = 0 in the general form (3.18); in other words all

the terms in the sequence are 1 :

ai = a 1 +d (i− 1)

= 1 + 0. (i− 1)

= 1

{ai} ={1; 1; 1; 1; 1; ...}

If we wish to sum this sequence from i = 1 to any positive integer n, we would write

�n

i=1

ai=

�n

i=1

1 = 1 + 1 + 1 +··· + 1 (n times)