find the sum of the numbers from 1 to 100. no calculators allowed! 1+2+3+4+5+6……..+98+99+100 =...

Find the sum of the numbers from 1 to 100.

No calculators allowed!

1+2+3+4+5+6……..+98+99+100 = ?????

Sigma Notation and

The Sum of Natural Numbers

Aims:To recognise and be able to use the sum of

natural numbers formula.To interpret sigma notation.

To apply these to more complex problems.

Sum of Natural Numbers• Thought to be first found by famous

mathematician Gauss when he was in school.Demonstration:

1

2

3

4

5

6

Sum of Natural Numbers Animation

Sum of Natural Numbers Formula

)1(21 nnSn

Use the formula to help find the sum of:1)The first 50 natural numbers2)The first 120 natural numbers3)The natural numbers between 50 and 100

inclusive.4)The natural numbers between 100 and 200

inclusive.5)The sum of the even numbers up to 100.

Sum of Natural Numbers Questions)1(2

1 nnSn

B

nn A

a

Last n value to use

First n valueto use

SIGMA means(SUM OF TERMS) Nth TERM

Formula

Sigma Notation Intro

j

4

1

j 2

21 2 2 3 2 24 18

7

4a

2a 42 2 5 2 6 72 44

nn 0

4

0.5 2

00.5 2 10.5 2 20.5 2 30.5 2 40.5 2

33.5

Sigma Notation Examples

Express each of the following as addition sums and find the total where possible.

1)

2)

3)

4

1r

r 4)

5)

6)

5

2

)1(r

r

5

3

2r

4

2

)1(2r

r

Sigma Notation Questions

Linking the two ideas

• Usually we will try to use the formula to help find the sum and not list all the terms.

• We can use the formula to help find the sum for any linear expression.

• Lets look at some examples.

Examples

50

1

5n

n

20

1

3n

n

n

25

1

32n

n

n

A few to try….

18

1

8n

n

30

1

14n

n

n

15

1

39n

n

n

15

1

20n

n

n

40

1

3100n

n

n

21

1

76n

n

n

Making things a bit harder

1)

2)

3)

50

7

)3)1(2(r

r

20

6

)23(r

r

100

21

)4(r

r

Plenary

a) Write the series in sigma notation:1, 4, 7, 10, 13, 16, ……..,1000.

b) Find the sum of the series.

c) Every third term of the above progression is removed, i.e. 7,16,…. Find the sum of the remaining terms.

Independent Study

Further Pure 1 Textbook:

Chapter 2 – Series - p14-17

Exercise 2A – p17