(c) find the sum of a sequence summation notation:read as “sigma” (ex) find the sum
TRANSCRIPT
(C) Find the Sum of a sequence
Summation notation: Read as “sigma”
1 2 31
...n
k nk
a a a a a
(ex) Find the sum.5
2
1k
k
Theorem on Sums:
Theorem on sum of a constant (c)
Theorem on sums
1 1 1
1 1 1
1 1
(1) ( )
(2) ( )
(3)
n n n
k k k kk k k
n n n
k k k kk k k
n n
k kk k
a b a b
a b a b
ca c a
for every real number c
1
(1)
(2) ( )
n
k
n
k m
c nc
c n m c
(12 – 2) Arithmetic sequences
Learning targets:
•To determine if a sequence is arithmetic
•To Find a formula for an arithmetic sequence
•To find the sum of an arithmetic sequence
Definition: An arithmetic sequence is defined recursively as
1 1, n na a a a d where d is the common difference.(A) I do (ex) Show that the sequence is arithmetic 4, 6, 8, 10,…
and d = 21 4a
n 1 2 3 4
an
2 2na n 1n na a d
0
1
2
3
4
2
2
2(1) 2
2(2) 2
2(3) 2
2(4) 2
2( ) 2n
d
a
a
a
a
a
a n
We do (ex) Show that the sequence if arithmetic if
3 2na n You do (ex) Show that the sequence if arithmetic if
6, 2, 2, ..., 4 10, ...n
<Try this> Work on #8 and 12 on page 821
(B) Find the particular term of an arithmetic sequence.
I do (ex) Find the forty-first tem of the arithmetic sequence:
2, 6, 10,14, 18, …
Step 1: Find
Step 2: Find using the formula.
Step 3: Find the required term.
1 ( 1)na a n d nth term of an arithmetic sequence is found by
1a
na
We do (ex) Find the term of the arithmetic sequence if
2 1a 18 49a 10aand find
Step 1: Find d
Step 2: find using the formula.
Step 3: Find using the formula.
Step 4: find the required term.
1a
na
You do (ex) Find the term of the arithmetic sequence if
8 75a 20 39a
naand Find
<Try this> #15, 27, and 29 on page 821
(C) Find the sum of an arithmetic sequence.
The sum of the first n term (nth partial sum of n term) Sn is given by:
or
1 2 3 ...n nS a a a a This means:
I do (ex) Find the sum: 60 + 64 + 68 + 72 + … + 120
Step 1: Find d : d = 4
Step 2: find the term of 120 using the formula.
Step 3: find the sum of the term using the formula.
1( )2n n
nS a a
1[2 ( 1) ]2n
nS a n d
We do (ex) Find the sum of all the even integers from through 100:The sequence is 2, 4, 6, …, 100, ...
Find an =
Find using the formula.nS
n 1 2 3 4 n
an 2 4 6 8
Sn
You do (ex) Find the partial sum of: 1 + 2 + 3 + 4 + …, + 100
<Try this> #35, 42, and 43 on page 822
n 1 2 3 4… 100
an
Sn
(12 – 3) geometric Sequences and Series
Learning targets:
• To show that a sequence is geometric
• To find a formula for geometric sequence
• To find the sum of a geometric sequence(A) Geometric Sequence
I do (ex) Find the nth term of the geometric sequence 2, 6, 18, 54, …, n
n 1 2 3 4 … n
an 2 6 18 54 …
Fill in the table.
Find the common ratio.
Write the recursive formula for the sequence.
Find the 10th term of the sequence.
1n
n
ar
a
11
nna a r
We do (ex) Find the nth term of the geometric sequence …, n
1 1 11, , , ,...
2 4 8
n 1 2 3 4 … n
an …
Fill in the table.
Find the common ratio.
Write the recursive formula for the sequence.
You do: (ex) Write the nth term of the geometric sequence.
1, -6, 36, -216, …
n 1 2 3 4 … n
an …
<try this> #25 and 28
(B) Geometric Series
(I) Finite geometric series
Finite sum (partial sum) Sn of a geometric sequence with the first term a1 and a common ration r ≠ 1 is
1
1
1
n
n
rS a
r
This means
n 1 2 3 4 n
an
Sn
2 3 11 1 1 1 1... n
nS a a r a r a r a r 2 1
3 2
4 3
5 4
a a r
a a r
a a r
a a r
We also use the summation notation (sigma)1
11
nk
nk
S a r
I do (ex) Find the sum of the first 7 terms of a geometric sequence.
-5, 15, -45, 135, …
n 1 2 3 4… n
an
Sn
Fill in the table, and find the common ratio.
Find the Sn1
1
1
n
n
rS a
r
11
nna a r
71
1
5( 3)K
K
We do (ex) Find the sum of the first 6 terms of a geometric sequence.1 1 1
1, , , ,...2 4 8
n 1 2 3 4… n
an
Sn
16
1
1
2
k
k
Find the sum using a graphing calculator.
2nd STAT MATH Sum(5)
2nd STAT OPS Seq (5)
½^(x-1),x,1,6,1)) ENTER
You do (ex) Find the sum of the first 8 terms of a geometric sequence.
2, -4, 8, -16, …n 1 2 3 4… n
an
Sn
(B-II) Infinite Geometric Series
Theorem:
If , then the infinite geometric series converges to the sum.1r
has the sum
1
1
aS
r
2 3 11 1 1 1 1... ...nS a a r a r a r a r
Otherwise, an infinite geometric series diverges.
I do (ex) Determine if the geometric series converges or diverges
4 82 ...
3 9
Find the common ratio.
Find the sum.
n 1 2 3 4… n
an
Sn
1
1
aS
r
If converges, find the sum.
We do (ex) Determine if the geometric series converges or diverges1
4 8 23 2 ...3
3 9 3
n
=
1
1
23
3
n
n
You do (ex) Determine if the geometric series converges or diverges
256 192 144 108 ...
<Try this> #51, 52, and 63 on page 831