Warm up

1. Warm up • 1. Find the sum of : • 2. Find the tenth term of the sequence if an = n2 +1:

2. Objective: To identify an arithmetic sequence and find specific terms in that sequence. Lesson 11-2 Arithmetic Sequences & Series

3. Arithmetic Sequences • Arithmetic sequences increase by a constant amount • an = an-1 + d d = common difference Example:3, 5, 7, 9, 11, 13, ... The terms have a common difference of 2. The common difference is the number d.

4. Example • Is the sequence arithmetic? –45, –30, –15, 0, 15, 30 • Yes, the common difference is 15

5. Finding any term in an Arithmetic Sequence • To find any term in an arithmetic sequence, use the formula • an = a1 + (n – 1)d • where d is the common difference. • Can also be used to find the number of terms in a finite arithmetic sequence.

6. Example • Find a formula for the nth term of the arithmetic sequence in which the common difference is 5 and the first term is 3. • an = a1 + (n – 1)d • a1 = 3 d = 5 • an = 3 + (n – 1)5

7. Example • If the common difference is 4 and the first term is -1, what is the 10th term of an arithmetic sequence? • an = a1 + (n – 1)d • d = 4 and a1 = -1 • a10= –1 + (10 – 1)4 • a10 = 35

8. Practice • If the first 3 terms in an arithmetic progression are 8,5,2 then what is the 16th term? In this progression a = 8 and d = -3. • an = a + (n - 1)d • a16 = 8 + (16 – 1)(-3) • = -37

9. Sum of an Arithmetic Series • To find the sum of an arithmetic series, we can use summation notation. • Which can be simplified to:

10. Example • Find the sum of the first 100 terms of the arithmetic sequence 1, 2, 3, 4, 5, 6, ... n = 100 = 5050

11. Practice • Find the sum of each series • 1. 3 + 6 + 9 + 12 + 15 + 18 + 21 + 24 + 27 = 135 • 2. 6 + 14 + 22 + 30 + …+ 54 = 210 • 3. 9 + 18 + 27 + 36 + 45 + 54 + 63 + 72 + 81 + 90 = 495