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SECTIONS 9-2 and 9-3 : ARITHMETIC &

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1 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

2 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, You multiply by 4 to get the next term. 3) 3, 12, 72, _______, _______, _______ SOLUTION : I don’t know either. There is no pattern.

3 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.

4 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.

5 Given the arithmetic sequence :
a1 a2 a3 a4 a5 a6 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

6 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

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

8 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

9 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 = = 506 Therefore, the 100th term is 506

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

11 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.

12 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 = n – 1 = n Therefore, there are 12 terms in the sequence.

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

14 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.

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

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

17 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.

18 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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