Four Rules
Notes
Suppose a1 , a2 , a3 , . . . and b1 , b2 , b3 , . . . are sequences and c is some
number c ∈ R.
k
k
X
X
1. Rule 1:
c · ai = c
ai
i=1
2. Rule 2:
3. Rule 3:
4. Rule 4:
k
X
i=1
k
X
i=1
k
X
i=1
ai +
ai −
k
X
i=1
k
X
bi =
bi =
i=1
k
X
(ai + bi )
i=1
k
X
(ai − bi )
i=1
c =k ·c
i=1
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Math 1050
1/9
Sum of first k terms in an arithmetic sequence
Notes
Gauss (late 18th century) found that 1 + 2 + · · · + 99 + 100 = 5050
quickly.
I a1 = 1, a2 = 2, a3 = 3, . . . is an arithmetic sequence with d = 1.
I a1 + a100 = 1 + 100 = 101, a2 + a99 = 2 + 99 = 101 and so on.
I Question: Do all arithmetic sequences have the pattern where
a1 + ak equals a2 + ak−1 equals a3 + ak−2 and so on? (That is,
the first term plus the last term equals the second term plus the
next-to-last term, and so on.)
To help you decide, try these:
1) What is the sum of the first 200 terms of the arithmetic sequence
8, 5, 2, −1, . . .?
2) What is the sum of the first 7 terms of the arithmetic sequence
14, 23, 32, 41, . . .?
(University of Utah)
Math 1050
2/9
Sum of first k terms in an arithmetic sequence
Notes
Sum of first k terms in an arithmetic sequence
If a1 , a2 , a3 , . . . is an arthmetic sequence, then an+1 = an + d for some
d ∈ R and the sum of the first k terms
k
X
i=1
ai =
k
(a1 + ak ).
2
Use the above formula for the following sums:
1) What is the sum of the first 26 terms of the sequence
−20, −15, −10, −5, . . .?
2) The sum of the first 60 terms of the sequence 73, 74, 75, 76, . . .
60
60
equals
(73 + 132) = (205) = 6150.
2
2
(University of Utah)
Math 1050
3/9
Infinite geometric sums
Notes
A special rule:
Infinite Geometric Sum
Suppose a1 , a2 , a3 , . . . is a geometric sequence where an+1 = r · an
where r ∈ R. Also suppose −1 < r < 1. Then
∞
X
ai =
i=1
a1
1−r
Example: The sum of the terms 1, 12 , 14 , 18 , . . . equals 2. This sequence
is geometric because we know its rule is built by finding r ∈ R. Its
rule is an+1 = 12 an and a1 = 1. Note r = 21 is between -1 and 1.
1+
1 1 1
1
+ + + ... =
2 4 8
1−
(University of Utah)
1
2
=
1
=2
( 12 )
Math 1050
4/9
Series
Notes
An infinite sum is called a series.
For many sequences a1 , a2 , a3 , . . ., the sum
∞
X
ai probably doesn’t
i=1
make sense.
Example of nonsense: The series for the sequence 1, 2, 3, 4, . . . does
not make sense. Why?
Example of nonsense: The series for the sequence 1, 3, 9, 27, . . . does
not make sense. Why?
Example of nonsense: The series for the sequence 1, −1, 1, −1, . . .
does not make sense. Can you convince yourself?
(University of Utah)
Math 1050
5/9
Series
Notes
A geometric sequence a1 , a2 , a3 , . . . with −1 < r < 1 has a series that
does make sense. Then
∞
X
i=1
ai =
a1
1−r
Example: Sum all of the terms of the geometric sequence
6, 4, 38 , 16
,...
9
(University of Utah)
Math 1050
6/9
More Exercises - Arithmetic sums
Notes
Example: Suppose your cable provider bills you for $20 for the
starting month, and that each month after that they increase your
monthly bill by $0.50. How much will you spend in the next three
years? (36 months)
(University of Utah)
Math 1050
7/9
More Exercises - Arithmetic sums
Notes
Example: Suppose you are offered a job (A) with a starting salary of
$32,000 with a raise of $2,000 per year. A competing job (B) offers
you a starting salary of $29,000 with a raise of $2,600 per year.
Which job will pay more over the first ten years? How much more?
(University of Utah)
Math 1050
8/9
More Exercises - Geometric Series
Notes
Example: A reservoir of water today collects 8 million gallons of new
water per year, but is losing 40% of the new water per year. How
much new water is collected until the reservoir cannot collect new
water?
(University of Utah)
Math 1050
9/9