A2HCh1102 Arithmetic Sequences and Partial Sums
Homework and Reading
Read p815 – 820
(1) HW p821#1– 47 odd
Goal
p1
Students recognize, write, and find the nth term of arithmetic sequence, and find the nth
partial sum of arithmetic sequences.
(2) HW p822 #57 – 91odd
Arithmetic Sequences
Example
Arithmetic Sequences
Def. of Arithmetic Sequences
A sequence is arithmetic if the difference between
consecutive terms are the same. So, the sequence
a1 , a2 , a3 , ... , an , ...
Find the common difference.
A 5, 8, 11, 14, ...
d = a2 ! a1 = 8 ! 5 = 3
d = a3 ! a2 = 11 ! 8 = 3
B
15, 11, 7, 3, ...
d = a2 ! a1 = 11 ! 15 = !4
d = a3 ! a2 = 7 ! 11 = !4
C
1, 4, 9, 16, ...
d = a2 ! a1 = 4 ! 1 = 3
d = a3 ! a2 = 9 ! 4 = 5
NOT ARITHMETIC
is arithmetic if there is a number d such that
a2 ! a1 = a3 ! a2 = a4 ! a3 = ! = d
The number d is the common difference of the
arithmetic sequences.
Arithmetic Sequence Formulas
Arithmetic Sequence Formulas
The n th term of an Arithmetic Sequence
The n th term of an arithmetic sequence has the form
an = dn + c
Linear Form
where d is the common difference between consecutive
terms of the sequence and c = a1 ! d. Substituting a1 ! d
for c in an = dn + c yields an alternative recursion form
for the n th term of an arithmetic sequence.
an = a1 + ( n ! 1) d
a2 = a1 + d
a3 = a2 + d = ( a1 + d ) + d = a1 + 2d
a4 = a3 + d = ( a1 + 2d ) + d = a1 + 3d
= a4 + d = ( a1 + 3d ) + d = a1 + 4d
a5!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!###################"
difference is 1
!
an = a1 + ( n ! 1) d
Explicit Form
Example
Find the n th term of the arithmetic sequence with
common difference 5 and first term 9.
a1 = 9
d=5
an = a1 + ( n ! 1) d
an = 9 + ( n ! 1) 5
an = 9 + 5n ! 5
an = 5n + 4
Example
The explicit form is developed by writing the sequence...
a1 = a1
Find the 32nd term of the sequence
34, 37, 40, 43, ...
37 ! 34 = 3 = d
a1 = 34
Example
Find the n th term of the arithmetic sequence
with fifth term 19 and ninth term 27.
a5 = 19 = a1 + 4d !
" ... solve system...
a9 = 27 = a1 + 8d #
d = 2, a1 = 11
an = a1 + ( n $ 1) d
an = 11 + ( n $ 1) 2
an = 11 + 2n $ 2
an = 2n + 9
Try :
Write the first five terms of the arithmetic sequence
A a4 = 16, a10 = 46
1, 6, 11, 16, 21
B
a8 = 23, a12 = 33
5.5, 8, 10.5, 13, 15.5
an = a1 + ( n ! 1) d
an = 34 + ( n ! 1) 3
an = 34 + 3n ! 3
an = 3n + 31
a32 = 3( 32 ) + 31
a32 = 127
Find the 32nd term of the sequence
C 101, 105, 109, ...
225
D
3, 1, ! 1, ! 3, ...
!59
A2HCh1102 Arithmetic Sequences and Partial Sums
The Sum of a Finite Arithmetic
Sequence
The Sum of a Finite Arithemic Sequence (series)
The Sum of a Finite Arithmetic Sequence
Example
Use the finite sequence 5, 9,13,17, 21, 25, 29.
Write the related
2nd W series. Evaluate the series.
ay
The Sum of a Finite Arithmetic Sequence
The sum of a finite arithmetic sequence with n term is
Sn =
You
ay als
the+se29
S = 5 + 9 +A13
+m17
+ o21ad+d 25
ries this way
rrange in orde
.
r, then add th
S = 119
e reverse
S = 5+ 9+
13 + 17 + 21
+ 25 + 29
S = 29 + 25 +
21 + 17 + 13 +
9+ 5
2S = 34 + 34
+ 34 + 34 + 34
+ 34 + 34
2S = 238 ( 34
• 7)
S = 119
order.
Example
Find the partial sum
20
! 7n + 1
n=1
a1 = 7 (1) + 1 = 8
a20 = 7 ( 20 ) + 1 = 141
20
! 7n + 1 = 202 ( 8 + 141) = 1490
n=1
C
1 + 2 + 3 + 4 + ... + 100
S = 100 (1 + 100 ) = 5050
2
100
D
! 7n
n = 51
p2
S = 50 ( 357 + 700 ) = 26425
2
n
(a + a )
2 1 n
Example
Find the sum of the finite sequence 5, 9,13,17, 21, 25, 29.
Sn = n ( a1 + an ); n = 7, a1 = 5, a7 = 29
2
S = 7 ( 5 + 29 ) = 7 ( 34 ) = 7 (17 ) = 119
2
2
Try :
Write the related series. Evaluate the series.
A 0.3, 0.6, 0.9, 1.2, 1.5, 1.8, 2.1, 2.4, 2.7, 3.0.
S = 10 ( 0.3 + 3.0 ) = 16.5
2
B
100, 225, 200, 175, 150, 125
100, 125, 150, 175, 200, 225
S = 6 (100 + 225 )
2
S = 975