Download EQ: What is the formula to find the sum of an arithmetic sequence?

Accel Precalc
Unit 4: Sequences & Series
Lesson #2: Arithmetic
Sequences and Partial Sums
EQ: What is the formula to find
the sum of an arithmetic sequence?
Recall:
Find the nth term of an Arithmetic
Sequence:
Arithmetic Sequence --- ordered list of
numbers with a common difference called d
• What is the Explicit Formula for an Arithmetic
Sequence? Can find ANY
an


a

d
n

1
1
= _________________


dn  a1  d
an = _________________
term in sequence
*Open Form
*Closed Form
Ex 1. Write the explicit formula for the
sequence 4, 7, 10, 13, …
*Open Form
an = 4+3n – 3
an = 3n + 1
Then find a13.
*Closed Form
a13 = 4+3(13 – 1)= 40
a13 = 3(13) +1= 40
 What is the Recursive Formula for an
Arithmetic Sequence?
Can only find the
NEXT term in
a

d
,
a

#
n 1
1
an = _________________
sequence
Ex 2. The eighth term Common
of an arithmetic
difference
sequence is 75, and theadded
twentieth
term is 39.
to PREVIOUS
term
Find the first term and the common
difference. Give the recursive formula for
this sequence.
a8 = 75 a20 = 39
an = a1 + d(n – 1)
39  75  20  8  3
75 = a1 - 3(8 – 1)
a1 = 96
d = -3
an = an - 1 - 3, a1 = 96
DG15 --- 10 minutes
Find the sum of a Finite Arithmetic Sequence
Generate Terms of an Arithmetic Sequence 2 Ways:
1st Way:
Repeated addition of d to the initial term
Sn  a1  a2  a3  ...  an2  an1  an
 a1   a1  d    a1  2d   ...  a1   n  1 d 
2nd way:
Repeated subtraction of d from the
nth term
Sn  an  an1  an2  ...  a3  a2  a1
 an  an  d   an  2d   ...  an  n  1d 
RECALL: How can you solve systems of equations?
Graphing
1. _________________________
Substitution
2. ________________________
We will use elimination to
Elimination
3. ________________________
derive the sum formula.
Add these two versions of Sn.
Sn  a1  a1  d   a1  2d   ...  a1  n  1d 
 Sn  an  an  d   an  2d   ...  an  n  1d 
a1  dn  d
an  dn  d
n a1  an 
a1 and an are both added “n” times
2
Ex 3. Find the sum of the odd integers 1 to 19.
The formula says we need1 to know n, the first
term, and the nth term.
19
Since we are asked to find the sum of the first
10
10 terms, n = ______
n a1  an  101  19 
Sn 

 100
2
2
Ex. 4
Find S20 for the sequence
a n  6n  1
20
6(1) – 1 = 6 – 1 = 5
6(20) – 1 = 120 – 1 = 119
S20
205  119 

 1240
2
• Finding the nth partial sum of an infinite
arithmetic sequence--- sum of the first
n terms of an infinite sequence
Ex 5. Find the 150th partial sum of the
arithmetic sequence 5, 16, 27, 38, 49, …
an  11n  6
1644
1505  1644 
Sn 
 123,675
2
Ex. 6 An auditorium has 20 rows of seats. There are 20 seats in the
first row, 21 seats in the second row, 22 seats in the third row, and so
on. How many tickets need to be sold to sell out this auditorium for an
upcoming show?
To find the sum of all 20 rows, we
need:
1
20
20
n = ____
d = ___
and a1 = _____
What is the explicit formula for
this problem?
an  20  1(n  1)
 n  19
Find the 20th term:
a20  19  20  39
Find the sum of the rows:
2020  39 
S 20 
 590
2
ASSIGNMENT:
Textbook p. 635 odds #1 – 43,
59 – 71, 83 – 87
Thurs--- Test Unit 3: Matrices
Fri --- DG 15 Unit 4 L1 & L2