This document summarizes different types of series: arithmetic series, geometric series, finite series, and infinite series. It provides the formulas to calculate the sum of a finite arithmetic series and geometric series. For an arithmetic series, the sum is equal to the mean of the first and last term multiplied by the number of terms. For a geometric series with a common ratio not equal to 1 and a non-zero divisor, the sum can be calculated using the provided formula. The document also explains that for an infinite geometric series, if the ratio is greater than or equal to 1, then the series has no solution. It provides an example of calculating the total distance traveled by a ball dropped from a height that bounces back to 80%

1. Summation fomulas
Presented by: Claire Gui

2. Arithmetic
series
Geometric
series
finite
finite
infinite

3. The sum of a finite Arithmetic Series
 finite arithmetic series
 the sum=the mean of the first and the nth term, multiplied
by the number of terms
)
2
( 1
n
naa
nS
+
=

4. Example
 If the bottom row has 21 logs and the top row has
15 logs.Each each row has one less log than the
row below it. How many logs are in the pile?
 n=row=21-15+1=7
 logs: 126
2
2115
7 =
+
× ）（

5. The sum of a finite Geometric series
 common ratio r≠1
 divisor≠0
)
1
1
(1
r
r
aS
n
n
−
−
=

6. Example
 Find n such that of the Geometric series
1+5+25+125+625+...

3906n =S
6n
3906
5-1
5-1
1
n
=
=× ）（

7. The sum of an infinite Geometric Series
 if |r| 1, the series has no solution≥
r
a
S
−
=
1
1
|

8. Example
 A ball is dropped from a height of 10 feet.Each time it
hits the ground, it bounces to 80% of the previous
height. Find the total distance traveld by the ball.
up&down up&down ...)8.0(102)8.0(10210d 2
+×+×+=
908010
8.01
)8.0(20
10 =+=
−
+=d