9-2: Arithmetic Sequences and
Partial Sums
Learning Target: I will be able to
recognize, write, and find the
nth terms of arithmetic
sequences; find the nth partial
sums of arithmetic sequences;
use arithmetic sequences to
model and solve real-life
problems.
Arithmetic Sequence:
- Has a common difference
between terms (-3,-1,1,3,5,…)
EX: Find the next 4 terms in the
arithmetic sequence -12,-1,10,…
d=-1-(-12)=11
𝑎4 = 10 + 11 = 21
𝑎5 = 21 + 11 = 32;
𝑎6 = 32 + 11 = 43;
𝑎7 = 43 + 11 = 54
Find the nth term of a sequence:
𝑎𝑛 = 𝑎1 + (𝑛 − 1)(𝑑)
EX: Find the 41st term in the
arithmetic sequence 11,4,-3,-10,..
𝑑 = 4 − 11 = −7
𝑎41 = 11 + (41 − 1)(−7)
= 11 + (40)(−7) = 11 − 280
= −269
*Write a calculator program to
find the nth term of an
arithmetic sequence.
EX: Find the first term in the
arithmetic sequence for which
𝑎44 = 229 and d=8.
𝑎𝑛 = 𝑎1 + (𝑛 − 1)(𝑑)
229 = 𝑎1 + (44 − 1)(8) →
229 = 𝑎1 + (43)(8) →
229 = 𝑎1 + 344 → 229 − 344 = 𝑎1
→ −115 = 𝑎1
Arithmetic Means – the terms
between any two nonconsecutive terms
5, 10, x, y, 25, 30
x and y would be the two
arithmetic means between 10
and 25.
EX: Write an arithmetic
sequence that has 6 arithmetic
means between -12 and 23.
-12, __, __, __, __, __, __, 23
𝑎𝑛 = 𝑎1 + (𝑛 − 1)(𝑑) →
23 = −12 + (8 − 1)(𝑑) →
23 = −12 + 7𝑑 → 35 = 7𝑑 → 5 = 𝑑
-12, -7, -2, 3, 8, 13, 18, 23
Arithmetic Series – the sum of a
set of terms in an arithmetic
sequence. For example,
1+2+3+4+5=15.
Sum of a Finite Arithmetic
Series:
𝑛
𝑆𝑛 = (𝑎1 + 𝑎𝑛 )
2
EX: Find the sum of the first 32
terms in the arithmetic series
-12-6-0+…+174
We first need to determine if the
32nd term is 174.
𝑎32 = −12 + (32 − 1)(6) = 174
Now apply the formula for the
sum of a finite series.
𝑆32
32
(−12 + 174) = 16(162)
=
2
= 2592
*Let’s write a program to find
the sum of a finite series.
EX: Arnold sells clothing in a
store at a local mall. His
commission on the sale of new
spring merchandise is $3.50 per
item costing at least $25. To
encourage an increase in sales,
the manager promises a $0.05
increase in commission for each
additional item sold. How many
items will Arnold need to sell
for his total commission to be at
least $1000?
𝑎1 = 3.50, 𝑎2 = 3.55, 𝑎3 = 3.60, 𝑎4 = 3.65, …
𝑛
𝑆𝑛 = (𝑎1 + 𝑎𝑛 ) →
2
𝑛
1000 = (3.5 + 𝑎𝑛 ) →
2
𝑛
1000 = (3.5 + [𝑎1 + (𝑛 − 1)𝑑 ]) →
2
𝑛
1000 = (3.5 + [3.5 + (𝑛 − 1)(. 05)])
2
→
𝑛
1000 = (3.5 + 3.5 + .05𝑛 − .05)
2
𝑛
→ 1000 = (6.95 + .05𝑛)
2
→ 2000 = 𝑛(6.95 + .05𝑛)
→ 2000 = 6.95𝑛 + .05𝑛2
→ .05𝑛2 + 6.95𝑛 − 2000 = 0
Solve the quadratic equation.
−6.95 ± √6.952 − 4(.05)(−2000)
𝑛=
2(.05)
= 142.23
EX: Find a formula in slopeintercept form for the nth term
of the arithmetic sequence
whose common difference is 5
and whose first term is -1.
𝑎𝑛 = 𝑎1 + (𝑛 − 1)(𝑑)
= −1 + (𝑛 − 1)(5)
= −1 + 5𝑛 − 5 =
5𝑛 − 6
Y=5x-6
EX: The eighth term of an
arithmetic sequence is 25, and
the 12th term is 41. Write the
first five terms of this sequence.
𝑎𝑛 = 𝑎1 + (𝑛 − 1)(𝑑)
25 = 𝑎1 + (8 − 1)(𝑑)
41 = 𝑎1 + (12 − 1)(𝑑)
After some simplifying, we can
create two equations in two
variable.
𝑎1 + 7𝑑 = 25
𝑎1 + 11𝑑 = 41
Now solve the system.
𝑎1 = −3, 𝑑 = 4
The first five terms are
-3, 1, 5, 9, 13.
EX: Find the tenth term of the
arithmetic sequence that begins
8,20,…
𝑎𝑛 = 𝑎1 + (𝑛 − 1)(𝑑) →
𝑎10 = 8 + (10 − 1)(12) = 8 + (9)(12)
= 8 + 108 = 116
We could also have used our
program.
EX: Find the sum of the first 7
terms of the sequence 40, 37, 34,
…
We first need to find the seventh
term.
𝑎7 = 40 + (7 − 1)(−3)
= 40 + (6)(−3) = 40 − 18 = 22
Now apply the formula for the
sum of a finite series.
7
7
𝑆7 = (40 + 22) = (62) = 217
2
2
Again, we could use our
program.
EX: Find the sum of the given
integers.
a.) 1 to 35
b.) 1 to 57
a.) 𝑆35 =
b.) 𝑆57 =
35
2
57
2
(1 + 35) = 630
(1 + 57) = 1653
EX: Find the 50th partial sum of
the arithmetic sequence
-6,-2,2,6,…
𝑎50 = −6 + (50 − 1)(4)
= −6 + (49)(4) = −6 + 196
= 190
𝑆50
50
(−6 + 190) = 25(184)
=
2
= 4600
EX: Determine the seating
capacity of an auditorium with
30 rows of seats if there are 20
seats in the first row, 22 seats in
the second row, 24 seats in the
third row, and so on.
Find the number of seats in the
last row.
𝑎30 = 20 + (30 − 1)(2)
= 20 + (29)(2) = 20 + 58 = 78
Now find the sum of the first 30
rows.
30
(20 + 78) = 15(98) = 1470
𝑆30 =
2
EX: Consider a job offer with a
starting salary of $32,500 and an
annual raise of $2,500.
Determine the total
compensation from the company
through six full years of
employment.
Find the amount earned during
the 6th year.
𝑎6 = 32500 + (6 − 1)(2500) = 45000
Now find the sum of the series.
6
𝑆6 = (32500 + 45000) = 232500
2
Upon completion of this lesson,
you should be able to:
1. Find the terms in an
arithmetic sequence.
2. find arithmetic means.
3. Find the sum of a finite
arithmetic sequence.
For more information, visit
http://www.mathsisfun.com/algebra/sequen
ces-sums-arithmetic.html
HW Pg.659 3-48 3rds, 57-72
3rds, 84, 105-110.