Chapter 1: Number Patterns1.3: Arithmetic SequencesEssential Question: What is the symbol for summation?

1.3 Arithmetic Sequences•Arithmetic Sequence → a sequence where

the difference between each term and the preceding term is constant (adding/subtracting the same number repeatedly)

•Example 1: Arithmetic Sequence▫Are the following sequence arithmetic?▫If so, what is the common difference?

a){ 14, 10, 6, 2, -2, -6, -10 }b){ 3, 5, 8, 12, 17 }

Yes; common difference is -4

Not arithmetic

1.3 Arithmetic Sequences• In an arithmetic sequence {un}

(Recursive Form): un = un-1 + dfor some constant d and all n ≥ 2

• Example 2: Graph of an Arithmetic Sequence▫If {un} is an arithmetic sequence with u1=3

and u2=4.5 as the first two terms

a)Find the common differenceb)Write the recursive functionc)Give the first seven terms of the sequenced)Graph the sequence

1.3 Arithmetic Sequences• Example 2: Graph of an Arithmetic Sequence

▫If {un} is an arithmetic sequence with u1=3 and u2=4.5 as the first two terms

a)Find the common difference

b)Write the recursive function

c)Give the first seven terms of the sequence

d)Graph the sequence

d = 4.5 – 3 = 1.5

u1 = 3 and un = un-1 + 1.5 for n ≥ 2

3, 4.5, 6, 7.5, 9, 10.5, 12

See page 22, figure 1.3-1b. Note that the graph is a straight line (this will become relevant later)

1.3 Arithmetic Sequences

•Explicit Form of an Arithmetic Sequence▫Arithmetic sequences can also be

expressed in a form in which a term of the sequence can be found based on its position in the sequence

•Example 3: Explicit Form of an Arithmetic Sequence▫Confirm that the sequence un = un-1 + 4

with u1=-7 can also be expressed as un = -7 + (n-1) ∙ 4

1.3 Arithmetic Sequences•Example 3, continued

▫u1 = -7

▫u2 = -7 + 4 = -3

▫u3 = (-7 + 4) + 4 = -7 + 2 ∙ 4 = -7 + 8 = 1

▫u4 = (-7 + 2 ∙ 4) + 4 = -7 + 3 ∙ 4 = -7 + 12 = 5

▫u5 = (-7 + 3 ∙ 4) + 4 = -7 + 4 ∙ 4 = -7 + 16 = 9

•Explicit Form of an Arithmetic Sequence▫un = u1 + (n-1)d for every n ≥ 1

u1 → 0 “4”s

u2 → 1 “4”

u3 → 2 “4”s

u4 → 3 “4”s

u5 → 4 “4”s

1.3 Arithmetic Sequences•Example 4

▫Find the nth term of an arithmetic sequence with first term -5 and common difference 3

•Solution▫Because u1 = -5 and d = 3, the formula would

be:un = u1 + (n-1)d un = = -5 + 3n – 3 =

•So what would be the 8th term in this sequence? The 14th? The 483rd?

-5 + (n-1)3

3n - 8

1.3 Arithmetic Sequences

•Formula: un = u1 + (n – 1)(d)•Example 5: Finding a Term of an

Arithmetic Sequence▫What is the 45th term of the arithmetic

sequence whose first three terms are 5, 9, and 13?

•Solution▫ ▫ ▫

u1 = 5The common difference, d, is 9-5 = 4u45 = u1 + (45 – 1)d = 5 + (44)(4) = 181

1.3 Arithmetic Sequences• Example 6: Finding Explicit and Recursive Formulas

▫ If {un} is an arithmetic sequence with u6=57 and u10=93, find u1, a recursive formula, and an explicit formula for un

• Solution:▫The sequence can be written as: …57, ---, ---, ---,

93, ... u6, u7, u8, u9, u10

▫The common difference, d, can be found like the slope93 57 36

910 6 4

d

1.3 Arithmetic Sequences• The value of u1 can be found using the explicit form of

an arithmetic sequence, working backwards:un = u1 + (n-1)du6 = u1 + (6-1)(9) We don’t know u1 , but we know u6

57 = u1 + 4557 – 45 = u1

12 = u1

• Now that we know u1 and d, the recursive form is given by:

u1 = 12 and un = un-1 + 9, for n ≥ 2• The explicit form of the arithmetic sequence is given by:

un = 12 + (n-1)9un = 12 + 9n – 9un = 9n + 3, for n ≥ 1

1.3 Arithmetic Sequences

•Today’s assignment:Page 29, 1-6, 25-30Ignore directions for graphing

•Tomorrow:Summation NotationCalculators will be important

Chapter 1: Number Patterns1.3: Arithmetic Sequences (Day 2)Essential Question: What is the symbol for summation?

1.3 Arithmetic Sequences

•Summation Notation▫When we want to find the sum of terms in a

sequence, we use the Greek letter sigma: ∑

▫Example 7

1 2 31

means ...m

k mk

c c c c c

5

1

( 7 3 ) ( 7 3 1) ( 7 3 2) ( 7 3 3)

( 7 3 4) ( 7 3 5)

( 4) ( 1) (2) (5) (8)

10

n

n

1.3 Arithmetic Sequences• Computing sums with a formula

▫Either of the following formulas will work:

▫What they mean:a) Add the first & last values of the sequence,

multiply by the number of times the sequence is run, divide by 2

b) Number of times sequence is run (k) multiplied by 1st value. Add that to k(k-1), divided by 2, multiplied by the common difference

11

11

) ( )2

( 1))

2

k

n kn

k

nn

ka u u u

k kb u ku d

Version A

•Using those formulas:•

50

1

11

1

50

50

1

7 3

) ( )2

7 3(1) 4

7 3(50) 143

507 3 (4 143) 3475

2

n

k

n kn

k

n

n

ka u u u

u

u u

n

Version B50

1

11

1

50

1

7 3

( 1))

2

7 3(1) 4

3

50(50 1)7 3 50(4) ( 3) 3475

2

n

k

nn

n

n

k kb u ku d

u

d

n

•

1.3 Arithmetic Sequences

•Computing sums with the TI-86▫Storing regularly used functions:

2nd, CATLG-VARS (CUSTOM), F1, F3 Move down to function desired, store with

F1-F5 We will need sum & seq( for this section

▫Using sequence: seq(function, variable, start, end)

▫Using sum: sum sequence

1.3 Arithmetic Sequences

seq (function, variable, start, end)sum seq(7-3x,x,1,50) = -3475

50

1

7 3n

n

1.3 Arithmetic Sequences• Partial Sums

▫The sum of the first k terms of a sequence is called the kth partial sum

• Example 9: Find the 12th partial sum of the arithmetic sequence: -8, -3, 2, 7, …▫Formula method:1

1

12

12

1

8

5

( 1)( )

8 (12 1)(5) 47

12( 8 47) 234

2

n

nn

u

d

u u n d

u

u

1.3 Arithmetic Sequences• Example 9: Find the 12th partial sum of the

arithmetic sequence: -8, -3, 2, 7, …▫Calculator method:

Need to determine equation:un=-8+(n-1)(5) [you can simplify] -8+5n-5=5n-13

sum seq(5x-13,x,1,12)

Assignment

•Page: 29 – 30•Problems: 7 – 17 & 31 – 43 (odd)

▫Write the problem down▫Make sure to show either

The formula you used when solving the sum; or

What you put into the calculator to find the answer