11.2 – Arithmetic Sequences

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

Important Formulas for Arithmetic Sequence: Recursive Formula Explicit Formula a1= an = an – 1 + d an = a1 + (n-1)d Where: an is the nth term in the sequence a1 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

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

Ex: Write the explicit and recursive formula for each sequence 2, 4, 6, 8, 10 First term: a1 = 2 difference = 2 an = a1 + (n-1)d a1 = an = 2 + (n-1)2 Explicit: an = an – 1 + 2, where a1 = 2 an = an – 1 + d Recursive:

Explicit Arithmetic Sequence Problem Find the 25th term in the sequence of 5, 11, 17, 23, 29 . . . an = a1 + (n-1)d Start with the explicit sequence formula Find the common difference between the values. difference = 6 a25 = 5 + (25 -1)6 Plug in known values a25 = 5 + (24)6 =149 Simplify

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? an = a1 + (n-1)d Start with the explicit sequence formula Find the common difference between the values. You will save $12 a week so this is your difference. difference = 12 a26 = 75 + (27 -1)12 Plug in known values 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. a26 = 75 + (26)12 =$387 Simplify

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

Explicit Arithmetic Sequence Problem Find the 25th term of the arithmetic sequence: 26, 13, 0, -13 an = a1 + (n-1)d Start with the explicit sequence formula Find the common difference between the values. difference = -13 a25 = 26 + (25 -1)(-13) Plug in known values a25 = 26 + (24)(-13) =-286 Simplify