1. Recursive vs. Explicit

2. Arithmetic Sequence – Geometric Sequence – Nth term – Recursive – Explicit –

3. Arithmetic Geometric  Sequences of numbers that follow a pattern of adding a fixed number from one term to the next are called arithmetic sequences.  Example: 15, 12, 9, 6,... 15 - 3 = 12 12 – 3 = 9 9 – 3 = 6  Geometric sequences follow a pattern of multiplying by a fixed number from one term to the next.  Example: 9, 18, 36, 72, … 9 x 2 = 18 18 x 2 = 36 36 x 2 = 72

4. Recursive A recursive equation is based on the changes happening to the value of the previous term. Explicit An explicit equation is based on what changes happen to the term number to get the sequence result.

5. Let’s look at the arithmetic sequence from before. 15, 12, 9, 6, … In math a subscript is like an adjective in English

6. Write a recursive equation for an arithmetic sequence from the toothpick activity. We’ll share answers in a few minutes.

7. Let’s try the geometric sequence! 9, 18, 36, 72,... Hey, that’s easy

8. Write a recursive equation for a geometric sequence from the toothpick activity. We’ll share answers in a few minutes.

9. While recursive equations are great for describing the pattern of a sequence, it’s not very good for finding the value of, say, the 100 th term. For this we need to look at how the term number relates to the sequence answer. Let’s look at the arithmetic sequence again, but this time as a table For the explicit formula we need to figure out how do we get from 1 to 15, 2 to 12, 3 to 9 and 4 to 6. Sometimes its helpful to look at the zero term. In this case term zero is 18. Term #1234 Value151296 Term #0123 Value1815129

10. Term # zero tells us what is added or subtracted after the term number is multiplied or divided. And, that difference between values we looked at in the recursive equation is the amount the term number is multiplied by. Term #01st2ndnth Value181512

11. Write an explicit equation for an arithmetic sequence from the toothpick activity. We’ll share answers in a few minutes.

12. Ok, so how does it work for geometric sequences? Term #1234 value9183612 Once again lets look at term zero. This time term zero is a multiplier. Value = 4.5 x ? We can see for the first term: 4.5 x 2 = 9 But the second term is: 4.5 x 4 = 18 Term #01st2ndnth value4.5918

13. Term #01st2ndnth value4.5918

16. Write an explicit equation for geometric sequence from the toothpick activity We’ll share answers in a few minutes.