SECTIONS 9-2 and 9-3 : ARITHMETIC & ALGEBRA II HONORS/GIFTED : SECTIONS 9-2 and 9-3 (Arithmetic and Geometric Sequences) ALGEBRA II H/G @ SECTIONS 9-2 and 9-3 : ARITHMETIC & GEOMETRIC SEQUENCES

What are the next three terms in each sequence? 1) 3, 12, 21, _______, _______, _______ SOLUTION : 30, 39, 48. You add 9 to get the next term. 2) 3, 12, 48, _______, _______, _______ SOLUTION : 192, 768, 3072. You multiply by 4 to get the next term. 3) 3, 12, 72, _______, _______, _______ SOLUTION : I don’t know either. There is no pattern.

PROGRESSING TO INFINITY ARITHMETIC SEQUENCE : A progression in which one term equals a constant (common difference) added to the preceding term. GEOMETRIC SEQUENCE : A progression in which one term equals a constant (ratio) multiplied by the preceding term.

Determine whether each sequence is arithmetic, geometric, or neither Determine whether each sequence is arithmetic, geometric, or neither. Explain your answer. 4) 4, 7, 10, … SOLUTION : Arithmetic. The common difference is 3. 5) 2, 6, 24, … SOLUTION : Neither. There is neither a common difference nor a ratio. 6) 3, -6, 12, … SOLUTION : Geometric. The ratio is -2.

Given the arithmetic sequence : a1 a2 a3 a4 a5 a6 3 10 17 24 31 38 … NOTE : a1 means 1st term, a2 the 2nd term, and so on. 7) What is the common difference? SOLUTION : 7 a1 = 3 + 7(0) a4 = 3 + 7(3) a2 = 3 + 7(1) a5 = 3 + 7(4) a3 = 3 + 7(2) a6 = 3 + 7(5) 8) an = ___________________ SOLUTION : an = a1 + (n – 1)d

ARITHMETIC SEQUENCE FORMULA an = a1 + (n – 1)d RED means the formula is to be written on your formulas page. an stands for the nth term a1 stands for the first term n is the term number d is the common difference

Given the geometric sequence : a1 a2 a3 a4 a5 a6 3 6 12 24 48 96 … 9) What is the ratio? SOLUTION : 2 a1 = 3 • 20 a4 = 3 • 23 a2 = 3 • 21 a5 = 3 • 24 a3 = 3 • 22 a6 = 3 • 25 10) an = ___________________ SOLUTION : an = a1 • rn-1

GEOMETRIC SEQUENCE FORMULA an = a1 • rn-1 RED means the formula is to be written on your formulas page. an stands for the nth term a1 stands for the first term n is the term number r is the ratio

11) Calculate a100 for the arithmetic sequence 11, 16, 21, 26, … SOLUTION : an = a1 + (n – 1)d a100 = 11 + (100 – 1)5 = 11 + (99)5 = 11 + 495 = 506 Therefore, the 100th term is 506

12) Calculate a100 for the geometric sequence with a1 = 40 and r = 1.05. SOLUTION : an = a1 • rn-1 a100 = 40 • 1.05100-1 = 5009.571727 Therefore, the 100th term of the sequence is 5009.571727

13) The number 68 is a term in the arithmetic sequence with a1 = 5 and d = 3. Which term is it? SOLUTION : an = a1 + (n – 1)d 68 = 5 + (n – 1)3 68 = 5 + 3n – 3 68 = 3n + 2 66 = 3n 22 = n Therefore, 68 is the 22nd term.

14) A geometric sequence has a1 = 17 and r = 2. If an = 34816, find n. SOLUTION : an = a1 • rn-1 34816 = 17 • 2n-1 2048 = 2n-1 log2048 = (n – 1)log2 log2048 = n – 1 log 2 11 = n – 1 12 = n Therefore, there are 12 terms in the sequence.

15) Calculate a45 for the arithmetic sequence 100, 94, 88, 82,… SOLUTION : an = a1 + (n – 1)d a45 = 100 + (45 – 1)(-6) = 100 + (44)(-6) = 100 + (-264) = -164 Therefore, the 45th term is -164.

MEAN : average ARITHMETIC or GEOMETRIC MEANS : between two numbers are the terms which form an arithmetic sequence or a geometric sequence between the two given terms.

16) Insert 4 arithmetic means between 37 and 52. SOLUTION : 37, _______, _______, _______, _______, 52 an = a1 + (n – 1)d 37 + 3 = 40 = 37 + (6 – 1)d 40 + 3 = 43 52 = 37 + 5d 43 + 3 = 46 15 = 5d 46 + 3 = 49 3 = d

17) Insert 2 geometric means between 52 and 73. SOLUTION : 52, _______, _______, 73 an = a1 • rn-1 52 • 1.12 = 58.24 = 52 • r4-1 58.24 • 1.12 = 65.23 73 = 52 • r3 1.4038 = r3 1.12 = r

18) Find a1 for the arithmetic sequence with a19 = 50 and a20 = 53. SOLUTION : an = a1 + (n – 1)d 53 = a1 + (20 – 1)(3) 53 = a1 + (19)(3) 53 = a1 + 57 -4 = a1 Therefore, the 1st term of the sequence is -4.

19) Calculate a1 for the geometric sequence with a9 = 200 and a10 = 220. SOLUTION : an = a1 • rn-1 220 = a1 • (1.1)10-1 220 = a1 1.19 93.3 = a1 Therefore, the 1st term of the sequence is 93.3.

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