Warm up – Warm up –

Arithmetic SequencesArithmetic SequencesIdentify the next three terms and the common difference for the following sequences:1.) 0, 5, 10, 15, 20…2.) 74, 67, 60, 53 …

4.)1 1 1, , 0, ...2 4 4

3.) 9, 17, 25, 33 … 41, 49, 57 d = 8

46, 39, 32 d = -7

25, 30, 35 d = 5

, ,1 3

12 4

1

4d

Arithmetic Sequence

Recursive Formula – a formula for sequence for which one or more previous terms are used to generate the next term

•Term Number – The position of the term in a sequence•Consecutive – Two numbers with a difference of 1

Must have an initial condition that tells where the sequence starts. A recursion formula tells how any term of the sequence relates to the preceding term.

Uses t (term), n (which term number), d (common difference) and you must know the pattern (first term).

Recursive Formulas : t t dn n-1

19, 14, 9, 4, . . .

Initial condition:  t1 = 19 ; d = -5

Recursive formula: 

To write the recursive formula substitute the correct common difference for d.Ex:

t t dn n-1

t

t tn n

1

1

19

5

19, 14, 9, 4, . . .Tell what term 1 is equal to.Then write formula substituting

the correct common difference for d only.

Example:

t

t tn n

1

1

19

5

t2 = t2-1 + (– 5) (we know n = 2 & d = -

5, plug them in)

t2 = t1 – 5 (we know t1 = 19, so we

plug it in)

t2 = 19 – 5

t2 = 14 (Looking at the example, the 2nd

term is 14!)

Find the second term19, 14, 9, 4, . . .

t t dn n-1

Write the recursive formula for each:1.5 , 8 , 11 , 14 , 17 , ...

2.26 , 31 , 36 , 41 , 46 , ...

3.20 , 18 , 16 , 14 , 12 , ...

n n

t

t t

1

1

5

3

n n

t

t t

1

1

26

5

n n

t

t t

1

1

20

2

The recursive formula can lead into the recursive function. The recursive function is another way to create a table of values related to a sequence. An Arithmetic Sequence Recursive Function is also known as a Linear Function.

The Recursive Function y = dx + t₀

d = common difference, t₀ = term zero x = term #, y = term value

Example: 19, 14, 9, 4, . . .

d = -5

t₀ = 24 (since 19 is term 1, term 0 would be t1 - d or 19 +5 = 24)

y = -5x + 24 (Substitute these values in to the function formula)

So the recursive function would be y = -5x + 24

The recursive function is another way to create a table of values related to a sequence.

The Recursive Function y = dx + t₀

d = common difference, t₀ = term zero x = term #, y = term valueTo create a table of values, you will use the term number as your x-value and the term value as your y-value.

ORTo create a table of values you can use the recursive function you create from the sequence and plug in values for x and solve for y.NOTE: Once you create a table, you can then graph the function on a coordinate plan.

1. Write a Recursive Function for the following sequence.

2. Create a table of values using the recursive function and Graph the function.

3. Find the 30th term of the sequence using the function.

4, 7, 10, 13, …

d = 3

t₀ = 1 (since 4 is term 1, term 0 would be t1 - d or 4 - 3 = 1)

y = 3x + 1 (Substitute these values in to the function formula)

So the recursive function would be y = 3x + 1

REMEMBER:The Recursive Function y = dx + t₀ x = term #, y = term value1.

2.

1. Write a Recursive Function for the following sequence.

2. Create a table of values using the recursive function and Graph the function.

3. Find the 30th term of the sequence using the function.

4, 7, 10, 13, …The recursive function is y = 3x + 1

Term #

x 1 2 3 4

Term Value

y 4 7 10 13

3.

1. Write a Recursive Function for the following sequence.

2. Create a table of values using the recursive function and Graph the function.

3. Find the 30th term of the sequence using the function.

4, 7, 10, 13, …

y = 3x + 1

y = 3(30) + 1 (to find the 30th term substitute 30 for x)

y = 90+ 1 (multiply)

y = 91 (add)So the 30th term would be 91

Write the recursive function for each:1. 5 , 8 , 11 , 14 , 17 , ...2. 26 , 31 , 36 , 41 , 46 , ...3.20 , 18 , 16 , 14 , 12 , ...Create a table using the recursive functions above and find the 30th term:1. 5 , 8 , 11 , 14 , 17 , ...2. 26 , 31 , 36 , 41 , 46 , ...3.20 , 18 , 16 , 14 , 12 , ...