1. Warm up – Arithmetic Sequences Identify the next three terms and the common difference for the following sequences: 1.)0, 5, 10, 15, 20… 2.)74, 67, 60, 53 … 4.) 3.)9, 17, 25, 33 … 41, 49, 57d = 8 46, 39, 32d = -7 25, 30, 35d = 5

3. Arithmetic Sequence

4. 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

5. 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 :

6. 19, 14, 9, 4,... Initial condition: t 1 = 19 ; d = -5 Recursive formula: To write the recursive formula substitute the correct common difference for d. Ex:

7. 19, 14, 9, 4,...  Tell what term 1 is equal to.  Then write formula substituting the correct common difference for d only. Example:

8. t 2 = t 2-1 + (– 5) (we know n = 2 & d = -5, plug them in) t 2 = t 1 – 5 (we know t 1 = 19, so we plug it in) t 2 = 19 – 5 t 2 = 14 (Looking at the example, the 2 nd term is 14!) Find the second term 19, 14, 9, 4,...

9. Write the recursive formula for each: 1.5, 8, 11, 14, 17,... 2.26, 31, 36, 41, 46,... 3.20, 18, 16, 14, 12,...

10. 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 Functiony = 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 t 1 - 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

11. The recursive function is another way to create a table of values related to a sequence. The Recursive Functiony = dx + t₀ d = common difference, t₀ = term zero x = term #, y = term value To create a table of values, you will use the term number as your x-value and the term value as your y-value. OR To 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.

12. 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 30 th term of the sequence using the function. 4, 7, 10, 13, … d = 3 t₀ = 1 (since 4 is term 1, term 0 would be t 1 - 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 Functiony = dx + t₀ x = term #, y = term value 1.

13. 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 30 th term of the sequence using the function. 4, 7, 10, 13, … The recursive function is y = 3x + 1 Term # x1234 Term Value y471013

14. 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 30 th term of the sequence using the function. 4, 7, 10, 13, … y = 3x + 1 y = 3(30) + 1 (to find the 30 th term substitute 30 for x) y = 90+ 1 (multiply) y = 91 (add) So the 30 th term would be 91

15. 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 30 th term: 1. 5, 8, 11, 14, 17,... 2. 26, 31, 36, 41, 46,... 3.20, 18, 16, 14, 12,...