IB Studies

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Adrian Sparrow

Arithmetic progressions:series and sequences 1

Arithmetic sequences

Find the next two terms of each of these sequences.

1. 3,5,7,9,11,

2. 7,11,15,19,23,

3. 1

2,1,

3

2,2,

5

2,

4. 5,4,3,2,1,

5. 25,20,15,10,5,

13,15

27,31

3,

7

2

0, 1

0, 5

These are arithmetic progressions as the difference between each is a constant added term.

Arithmetic sequences finding d

Find the difference (d) between each term.

1. 3,5,7,9,11,

2. 7,11,15,19,23,

3. 1

2,1,

3

2,2,

5

2,

4. 5,4,3,2,1,

5. 25,20,15,10,5,

d 2

d 4

d

1

2

d 1

d 5

Formula - examples

u

nu

1 (n 1)d

The nth term

The first termThe common

difference

Example 1

4,7,10,13,16,.....

Example 2

15,11,7,3, 1,.....

u

14, d 3

u1

15, d 4

u

n4 3(n 1)

u

n3n 1

u

n15 4(n 1)

u

n19n 4

Formula

1. 3,5,7,9,11,

2. 7,11,15,19,23,

3. 1

2,1,

3

2,2,

5

2,

4. 5,4,3,2,1,

5. 25,20,15,10,5,

Find the formula for the next term of each sequence.

u

n2n 1

u

n4n 3

u

n

1

2n

u

n6 n

u

n30 5n

Sums of arithmetic progressions

6 9 12 15 18 21

As the numbers form an arithmetic progression and now they are being added they become an arithmetic series.

S

n

n

22u

1 n 1 d or S

n

n

2u

1 u

n

The sum of n terms

The first term

The common difference

The sum of n terms

The first term

The last term

Formula - examples

S

n

n

22u

1 n 1 d or S

n

n

2u

1 u

n

Example 1

Find,

1+3+5+7+9+11+13

Example 2

Find the sum of the first

20 terms of,

10 12 14 16 ....

S

n

n

22u

1 n 1 d

Sn

n

2u

1 u

n u

11,u

n13, n 7

S

7

7

21 13

S

749

S

20

20

22(10) 19 2

u

110, d 2, n 20

S

20580

Sums

1. 3 5 7 9 ...

2. 7+11+15+19+...

3. 1

21

3

22 ...

4. 5+4+3+2+...

5. 25 20 15 10 ...

Find the sum of the first 30 terms of each of these series.

S

30960

S

301950

S

30232.5

S

30 285

S

30 1425