Aim: What are the arithmetic series and geometric series?

Do Now: Find the sum of each of the following sequences:

a) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19b) 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + . . . + 98 + 99 + 100

HW: p.265 # 12,14,20 p.272 # 6,8,10,16 p.278 # 8,10

The sum of an arithmetic sequence is called arithmetic series

Although we can find the arithmetic series one after the other, there is a formula to find the series faster.

2

)( 1 nn

aanS

5050)101(502

)1001(100100

S

Find the sum of the first 150 terms of the arithmetic sequence 5, 16, 27, 38, 49, . . .

First we need to determine what the last term of the 150 terms (or the 150th term) is.

a1 = 5; d = 16 – 5 = 11.

a150 = a1 + d(150 – 1), a150 = 5 + 11(149) = 1644

123675)1649(752

)16445(150150

S

Write the sum of the first 15 terms of the arithmetic series 1 + 4 + 7 + · · · in sigma notation and then find the sum

First of all, we need to find the recursive formula

),1(31 nan

a1 = 1 and d = 3

23 nan

15

1

23n

n

330)22(152

)44(15

43421)14(3115 aTo find the sum, we need to find a15

2

)431(1515

S

The sum of an geometric sequence is called geometric series

3, 15, 75, 375, 1875, 9375, 46875, 234375, 1171875 is a geometric sequence, find the sum of sequence.

The formula to find the finite (limited number of terms) geometric sequence is

r

raS

n

n

1

)1(1

4

)19531251(3

51

)51(3 9

9

S53

15r

14648434

)1953124(3

Write as a series and then find the sumn

n

)2

1(1010

5

1

16

5

8

5

4

5

2

5510

2

1r

21

1

21

1106

6

S

21

)6463(10

21641

110

16

315

32

630

2164630

Infinite series : The last term of the series is the infinity

An infinite arithmetic series has no limit

An infinite geometric series has no limit when

An infinite geometric series has a finite limit when the limit can be found by the formula

1r

1r

r

aSn

1

1

Find the sum of the following infinite geometric sequence: 4, 4(0.6), 4(0.6)2, 4(0.6)3, . . ., 4(0.6)n - 1 , . . .

a1 = 4 and r = 0.6 104.0

4

6.01

4

S

Find n

n

0 3

12 ,21 a

3

1r

3

322

31

1

2

S

Find the sum of the first 10 terms of the geometric

series .2

112

256

102310 S

Find the sum of five terms of the geometric series whose first term is 2 and fifth term is 162

S5 = 242