Transcript Homework

13.3 Arithmetic & Geometric
Series
A series is the sum of the terms of a sequence.
A series can be finite or infinite.
We often utilize sigma notation to denote a series
 is the greek letter sigma – stands for “sum”
n
a
k 1
index
k
 a1  a2  a3  ...  an
sequence rule
Ex 1) Express 3 – 6 + 9 – 12 + 15 using sigma notation
5
- five terms
k 1
(

1)
3k

- alternating signs
k 1
- rule 3k
Ex 2) Find the following sums.
j
a) 4  1  1 1 1 1 15
b) 7 2
  2 
j 1

   
2 4 8 16 16
t
t 0
 0  1  4  9  16  25  36  49
= 140

a
k 1
k
 a1  a2  a3  ... is an infinite series.
The sum of the first n terms is called the nth partial sum of the series and
is denoted by Sn.
Ex 3) Find the indicated partial sum.
a) S10 for –3 – 6 – 9 – 12 – 15 – 18 – 21 – 24 – 27 – 30 = –165
keep going…

b) S6 for
 (10m  6) = 4 + 14 + 24 + 34 + 44 + 54 = 174
m 1
Writing out all these terms is cumbersome! We have formulas!
If a1, a2, a3, … is an arithmetic sequence with common difference d
n(a1  an )
n(2a1  (n  1)d )
an = a1 + (n – 1)d
Sn 
or Sn 
2
Which should you use? Discuss advantages of each!
2
Ex 4) Find the indicated partial sum.
a) S8 for 15, 9, 3, –3, …
use
Sn 
8(2(15)  (8  1)(6))
S8 
 48
2

b) S24 for
 (1.5t  6)
t 1
n(a1  an )
use Sn 
2
n(2a1  (n  1)d )
2
24(4.5  30)
S24 
2
= 306
We can also use a formula for the sum of a geometric series.
If a1, a2, a3, … is a geometric sequence with common ratio r
n
n–1
a
(1

r
)
an = a1r
S  1
, r 1
n
1 r
Ex 5) Find the partial sum S7 for the series 1 – 0.8 + 0.64 – 0.512 + …
0.64
r
 0.8
0.8
1(1  (0.8)7 )
S7 
 0.672064
1  (0.8)
Be careful!
Watch order
of operations!
Ex 6) Marc’s grandmother gives him $100 on his birthday every year
beginning with his third birthday. It is deposited in an account that earns
7.5% interest compounded annually.
a1 = 100
r = 1.075 (why the 1??)
How much is the account worth the day after Marc’s 10th birthday?
100(1  1.0758 )
S8 
 $1044.64
1  1.075
Homework
#1303 Pg 695 #1–11odd, 17–18, 22, 24, 25, 27, 32, 36, 37