Everything Maths Grade 12

CHAPTER 3. SEQUENCES AND SERIES 3.6

Because all the terms are equal to 1 , it means that if we sumto n we will be adding n-number of 1 ’s
together, which is simply equal to n:

�n

i=1

1 = n (3.19)

Another simple arithmetic sequence is when a 1 = 1 and d = 1, which is the sequenceof positive
integers:

ai = a 1 +d (i− 1)

= 1 + 1. (i− 1)

= i

{ai} ={1; 2; 3; 4; 5; ...}

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

�n

i=1

i = 1 + 2 + 3 +··· +n (3.20)

This is an equation witha very important solution as it gives the answer to the sum of positive integers.

FACT

Mathematician, Karl

Friedrich Gauss, discov-

ered this proof when

he was only 8 years

old. His teacher had

decided to give his

class a problem which

would distract them

for the entire day by

asking them to add

all the numbers from

1 to 100. Young Karl

realised how to do this

almost instantaneously

and shocked the teacher

with the correct answer,

5050.

We first write Snas a sum of terms in ascending order:

Sn= 1 + 2 +··· + (n− 1) +n (3.21)

We then write the samesum but with the termsin descending order:

Sn= n + (n− 1) +··· + 2 + 1 (3.22)

We then add corresponding pairs of terms fromEquations (3.21) and (3.22), and we find that thesum
for each pair is the same, (n + 1):

2 Sn= (n + 1) + (n + 1) +··· + (n + 1) + (n + 1) (3.23)

We then have n-number of (n + 1)-terms, and by simplifying we arrive at the final result:

2 Sn = n (n + 1)

Sn =

n

2

(n + 1)

Sn=

�n

i=1

i =

n

2

(n + 1) (3.24)

Note that this is an example of a quadratic sequence.

General Formula for a Finite Arithmetic Se-

ries

EMCY

If we wish to sum any arithmetic sequence, thereis no need to work it out term-for-term. We willnow
determine the general formula to evaluate a finite arithmetic series. Westart with the general formula