Introduction to Arithmetic Sequences 18 May 2011.

Introduction to Arithmetic Sequences

18 May 2011

Arithmetic Sequences

When the difference between any two numbers is the same constant value

This difference is called d or the constant difference {4, 5, 7, 10, 14, 19, …} {7, 11, 15, 19, 23, ...}

← Not an Arithmetic Sequence

← Arithmetic Sequence

d = 4

Your Turn:

Determine if the following sequences are arithmetic sequences. If so, find d (the constant difference). {14, 10, 6, 2, –2, …} {3, 5, 8, 12, 17, …} {33, 27, 21, 16, 11,…} {4, 10, 16, 22, 28, …}

Recursive Form

The recursive form of a sequence tell you the relationship between any two sequential (in order) terms.

un = un–1 + d n ≥ 2

common difference

Writing Arithmetic Sequences in Recursive Form

If given a term and d

1. Substitute d into the recursive formula

Examples: Write the recursive form and find the next 3 terms

u1 = 39, d = 5 3

1d,

5

3u1

Your Turn: Write the recursive form and find the next 3 terms

u1 = 8, d = –2 u1 = –9.2, d = 0.9

Writing Arithmetic Sequences in Recursive Form, cont.

If given two, non-sequential terms

1. Solve for d

d = difference in the value of the terms

difference in the number of terms

2. Substitute d into the recursive formula

Example #1

Find the recursive formula u3 = 13 and u7 = 37

Example #2

Find the recursive formula u2 = –5 and u7 = 30

Example #3

Find the recursive formula u4 = –43 and u6 = –61

Your TurnFind the recursive formula:

1. u3 = 53 and u5 = 71 2. u2 = -7 and u5 = 8

3. u3 = 1 and u7 = -43

Explicit Form

The explicit form of a sequence tell you the relationship between the 1st term and any other term.

un = u1 + (n – 1)d n ≥ 1

common difference

Summary: Recursive Form vs. Explicit Form

Recursive Form

un = un–1 + d n ≥ 2

Sequential Terms

Explicit Form

un = u1 + (n – 1)d n ≥ 1

1st Term and Any Other Term

Writing Arithmetic Sequences in Explicit Form

You need to know u1 and d!!! Substitute the values into the explicit formula

1. u1 = 5 and d = 2 2. u1 = -4 and d = 5

Writing Arithmetic Sequences in Explicit Form, cont. You may need to solve for u1 and/or d.

1. Solve for d if necessary

2. Back solve for u1 using the explicit formula

u4 = 12 and d = 2

Example #2

u7 = -8 and d = 3

Example #3

u6 = 57 and u10 = 93

Example #4

u2 = -37 and u7 = -22

Your Turn:

Find the explicit formulas:

1. u5 = -2 and d = -6 2. u11 = 118 and d = 13

3. u3 = 17 and u8 = 92 4. u2 = 77 and u5 = -34

Using Explicit Form to Find Terms Just substitute values into the formula!

u1 = 5, d = 2, find u5

Using Explicit Form to Find Terms, cont.

u1 = -4, d = 5, find u10

Your Turn:

1. u1 = 4, d = ¼ 2. u1 = -6, d = ⅔

Find u8 Find u4

3. u1 = 10, d = -½ 4. u1 = π, d = 2

Find u12 Find u27

Summations Summation – the sum of the terms in a

sequence

{2, 4, 6, 8} → 2 + 4 + 6 + 8 = 20

Represented by a capital Sigma

Summation Notation

k

1nnuSigma

(Summation Symbol)

Upper Bound (Ending Term #)

Lower Bound (Starting Term #)

Sequence

Example #1

4

1nn2

Example #2

3

1n)3n(

Example #3

)2n3(3

1n

Your Turn: Find the sum:

5

1n)7n3(

4

1n)n45(


5

1n)n37(

4

1n]4)1n(3[


5

1n

2 )n30(

4

1n)2n(n

Partial Sums of Arithmetic Sequences – Formula #1

Good to use when you know the 1st term AND the last term

k

1nk1n )uu(

2

ku

# of terms

1st term last term

Formula #1 – Example #1Find the partial sum:

k = 9, u1 = 6, u9 = –24


k = 6, u1 = – 4, u6 = 14


k = 10, u1 = 0, u10 = 30

Your Turn:

Find the partial sum:

1. k = 8, u1 = 7, u8 = 42

2. k = 5, u1 = –21, u5 = 11

3. k = 6, u1 = 16, u6 = –19

Partial Sums of Arithmetic Sequences – Formula #2

Good to use when you know the 1st term, the # of terms AND the common difference

k

1n1n d

2

)1k(kkuu

# of terms

1st term common difference


k = 12, u1 = –8, d = 5


k = 6, u1 = 2, d = 5


k = 7, u1 = ¾, d = –½

Your Turn:

Find the partial sum:

1. k = 4, u1 = 39, d = 10

2. k = 5, u1 = 22, d = 6

3. k = 7, u1 = 6, d = 5

Choosing the Right Partial Sum Formula

Do you have the last term or the constant difference?

k

1n1n d

2

)1k(kkuu

k

1nk1n )uu(

2

ku

Examples Identify the correct partial sum formula:

1. k = 6, u1 = 10, d = –3

2. k = 12, u1 = 4, u12 = 100

Your Turn: Identify the correct partial sum formula

and solve for the partial sum

1. k = 11, u1 = 10, d = 2

2. k = 10, u1 = 4, u10 = 22

3. k = 16, u1 = 20, d = 7

4. k = 15, u1 = 20, d = 10

5. k = 13, u1 = –18, u13 = –102

Introduction to Arithmetic Sequences 18 May 2011.

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