1. 11.2 – Arithmetic Sequences

2. How do I know if it is an arithmetic sequence? Look for a common difference between consecutive terms Ex: 2, 4, 8, 16...Common Difference? 2+2 = 4 4+ 4 = 8 8 + 8 = 16 No. The sequence is NOT Arithmetic Ex: 48, 45, 42, 39...Common Difference? 48 - 3 = 45 45 - 3 = 42 42 - 3 = 39 Yes. The sequence IS Arithmetic. Subtract 3 each time

3. And Now.. Math Theory YUCH!

4. Important Formulas for Arithmetic Sequence: Recursive FormulaExplicit Formula a n = a n – 1 + d a n = a 1 + (n-1)d Where: a n is the nth term in the sequence a 1 is the first term n is the number of the term d is the common difference Arithmetic Mean The average between any arithmetic two values

5. How does this work: Start with the easy one: Find the missing term of the arithmetic sequence 84, ____, 110 Strategy: Find the average 84 + 110 = 97 2

6. Ex: Write the explicit and recursive formula for each sequence 2, 4, 6, 8, 10 First term: a 1 = 2 difference = 2 a n = a n – 1 + d a n = a n – 1 + 2, where a 1 = 2 Recursive: a n = a 1 + (n-1)dan = 2 + (n-1)2 Explicit:

7. Explicit Arithmetic Sequence Problem Find the 25 th term in the sequence of 5, 11, 17, 23, 29... a n = a 1 + (n-1)d a 25 = 5 + (24)6 =149 difference = 6 a 25 = 5 + (25 -1)6 Start with the explicit sequence formula Find the common difference between the values. Plug in known values Simplify

8. Explicit Arithmetic Sequence Problem Suppose you have saved $75 towards the purchase of a new iPad. You plan to save at least $12 from mowing your neighbor’s yard each week. In all, what is the minimum amount of money you will have in 26 weeks? a n = a 1 + (n-1)d a 26 = 75 + (26)12 =$387 difference = 12 a 26 = 75 + (27 -1)12 Start with the explicit sequence formula Find the common difference between the values. You will save $12 a week so this is your difference. Plug in known values Simplify WAIT: Why 27 and not 26 for n ? The first term in the sequence, 75, came before the weeks started (think of it as week 0). Therefore you want one more week in your formula to account for the $75 that you had before you started saving.

9. Let’s try some: Find the 25 th term of the arithmetic sequence: 26, 13, 0, -13

10. Explicit Arithmetic Sequence Problem Find the 25 th term of the arithmetic sequence: 26, 13, 0, -13 a n = a 1 + (n-1)d a 25 = 26 + (24)(-13) =-286 difference = -13 a 25 = 26 + (25 -1)(-13) Start with the explicit sequence formula Find the common difference between the values. Plug in known values Simplify