10 2 Arithmetic Sequences Date Arithmetic Sequence Sequence

10. 2 Arithmetic Sequences Date: ______

Arithmetic Sequence • Sequence in which each term after the first is obtained by adding a fixed number, called the difference, to the previous term. +3 +3 Common difference is 3. 5, 8, 11, 14, 17, . . . -2 -2 (d = 3) Common difference is -2. 16, 14, 12, 10, 8, . . . (d = -2)

Decide if each sequence is an arithmetic sequence. If yes, find the common difference. -5, -1, 3, 7, 11, . . . Yes. d = 4 4, 5, 7, 10, 14, … No. 1, 4, 8, 12, 16, … No. -4, -7, -10, -13, -16, … Yes. d = -3

Arithmetic Sequence an = a 1 + d(n − 1) an = nth term of the sequence a 1 = first term n = # of terms d = common difference

Find an and a 20. a 1 = 7 d=5 an = a 1 + d(n − 1) an = 7 + 5 n – 5 an = 2 + 5 n a 20 = 2 + 5(20) a 20 = 102

Find an and a 25. 48, 53, 58, 63, … an = a 1 + d(n − 1) 48 5 an = 48 + 5(n – 1) an = 48 + 5 n – 5 an = 43 + 5 n a 25= 43 + 5(25) a 25 = 168

Find an and a 25. -21, -39, -57, -75, … an = a 1 + d(n − 1) -21 -18 an = -21 – 18(n – 1) an = -21 – 18 n + 18 an = -3 – 18 n a 25= -3 – 18(25) a 25 = -453

Find an and a 20. a 17 = 22 d = -4 an = a 1 + d(n − 1) 22 = a 1 – 4(16) an = 86 – 4(n − 1) an = 86 – 4 n +4 an = 90 – 4 n 22 = a 1 – 64 86 = a 1 a 20 = 90 – 4(20) a 20 = 10 22 = a 1 – 4(17 − 1)

Find an and a 13. a 15 = 10 a 20 = 25 an = a 1 + d(n − 1) 10 = a 1 + 3(15 − 1) 10 = a 1 + 3(14) 10 = a 1 + 42 -32 = a 1 25 – 10 15 d= = =3 20 – 15 5 an = -32 + 3(n − 1) an = -32 + 3 n – 3 an = -35 + 3 n a 13 = -35 + 3(13) a 13 = 4

Find an and a 13. a 12 = -23 a 27 = 37 an = a 1 + d(n − 1) -23 = a 1 + 4(12 − 1) -23 = a 1 + 4(11) -23 = a 1 + 44 -67 = a 1 37 − ‾ 23 60 d= = =4 27 – 12 15 an = -67 + 4(n − 1) an = -67 + 4 n – 4 an = -71 + 4 n a 13 = -71 + 4(13) a 13 = -19

Sum of a Finite Arithmetic Sequence ( ) Find the sum of the first 10 terms of the sequence if a 1 = -16 and a 10 = 20 ( ) S 10 = 20

Find the sum of the first 42 terms of the sequence if a 1 = 7 and a 42 = 239 ( ) S 42 = 5166

Find the sum of the first 100 terms of the sequence if a 1 = 5 and d = 3. ( ( S 100 = 15, 350 ) ) an = a 1 + d(n − 1) a 100 = 5 + 3(100 − 1) a 100 = 302

Find the sum of the first 24 terms of the sequence if a 1 = -4 and d = -6. ( ( S 24 = -1752 ) ) an = a 1 + d(n − 1) a 24 = -4 – 6(24 − 1) a 24 = -142

Find the sum of the first 50 terms of the sequence 34, 45, 56, 67, 78, … ( ( S 50 = 15, 175 ) ) an = a 1 + d(n − 1) a 50 = 34 + 11(50 − 1) a 50 = 573

Find the sum of the first 20 terms of the sequence 12, 18, 24, 30, 36, … ( ( S 20 = 1380 ) ) an = a 1 + d(n − 1) a 20 = 12 + 6(20 − 1) a 20 = 126