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(

Your Turn: Find the sum:

5

1n)n37(

4

1n]4)1n(3[

Your Turn: Find the sum:

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

Formula #1 – Example #2Find the partial sum:

k = 6, u1 = – 4, u6 = 14

Formula #1 – Example #3Find the partial sum:

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

Formula #2 – Example #1Find the partial sum:

k = 12, u1 = –8, d = 5

Formula #2 – Example #2Find the partial sum:

k = 6, u1 = 2, d = 5

Formula #2 – Example #3Find the partial sum:

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