9.4 Arithmetic Series
series- indicated sum of the terms in a sequence
finite series- has a first and last term
6+9+12+15+18 (sum is 60)
infinite series- continues without end
3+7+11+15+… (cannot find sum)
arithmetic sequence- a series whose terms form an arithmetic sequence (see examples above)
𝑛
The sum Sn of a finite arithmetic series a1+a2+a3+…+an: Sn=2 (a1+an)
a1=1st term
an=last term
n= number of terms
How to find the number of terms in a series:
1) Subtract last & first number
2) Divide by common difference
3) Add 1 for the first term
limits- the least and greatest values of a series
summation notation(Σ=sigma Greek letter for sum)
Example:
𝑢𝑝𝑝𝑒𝑟 𝑙𝑖𝑚𝑖𝑡
108
∑
𝑒𝑥𝑝𝑙𝑖𝑐𝑖𝑡 𝑓𝑜𝑟𝑚𝑢𝑙𝑎
The series ends with 108th term
∑𝑛+5
𝑙𝑜𝑤𝑒𝑟 𝑙𝑖𝑚𝑖𝑡
𝑛=3
The simplified explicit formula for each term is n+5
The series starts with 3rd term
How to write a series in summation notation:
1) Use explicit formula for arithmetic sequence
2) Substitute first term and common difference
3) Combine like terms & simplify
4) Use explicit formula with last term as an
5) Solve for n (upper limit)
How to use summation notation:
1) Use explicit formula for arithmetic sequence
2) Find a1 by using simplified expression (use 1 for n)
**use what n= if it is NOT 1 for “first term”
3) Find an by using simplified expression (use top number for n)
4) Use sum equation to find end sum
**use top number — bottom number + 1 for # of terms
Examples: Find the sum of each finite arithmetic series.
1) 4 + 7 + 10 + 13 + 16 + 19 + 22
2) 8 + 9 + 10 + … + 15
Write each arithmetic series in summation notation:
3) 4 + 8 + 12 + 16 + 20
4) (-3) + (-6) + (-9) + … + (-30)
Find the sum of each finite series.
5)
6)
7)
Arena:
8) There are 30 rows of seats in a large arena. The first row contains 10 seats. Each successive row increases by 3 seats.
How many seats are in the last row? How many seats in all?
9.5 Geometric Series
Geometric series- the sum of the terms of a geometric sequence
Sum of a Finite Geometric Series:
𝑎1 (1 − 𝑟 𝑛 )
𝑆𝑛 =
1−𝑟
a1=1st term
an=nth term
n=number of terms
r=common ratio
*Use explicit formula (an=a1•rn-1) to simplify expression
*To find number of terms:
-given a series:
use what you know about exponents(or log) to solve for n
-given summation notation:
ends with 5th term
top number – bottom number +1
𝑢𝑝𝑝𝑒𝑟 𝑙𝑖𝑚𝑖𝑡
∑
5
𝑒𝑥𝑝𝑙𝑖𝑐𝑖𝑡 𝑓𝑜𝑟𝑚𝑢𝑙𝑎
𝑙𝑜𝑤𝑒𝑟 𝑙𝑖𝑚𝑖𝑡
1st term outside;
common ratio
always inside
𝑛−1 parentheses
∑ 5(3)
𝑛=1
*if missing, it =1
starts with 1st term
Examples:
What is the sum of the finite geometric sequence?
1)
2)
-15 + 30 – 60 + 120 – 240 + 480
10
∑ 5(−2)𝑛−1
𝑛=1
3)
4)
3 + 6 + 12 + 24 + … + 3072
11
∑ 3(−1.5)𝑛
𝑛=0
5)
6
∑ (−2)𝑛−1
𝑛=2