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Geometric and arithmetic sequences
Recursive AND Explicit Forms
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Definitions Arithmetic Sequences Geometric Sequences
Goes from one term to the next by adding and subtracting. Geometric Sequences Goes from one term to the next by multiplying or dividing.
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Arithmetic Sequences Recursive Form: an = an-1 + d
Explicit Form: an = a + (n – 1)d “an ” means the n-th term (what you are looking for) “an-1 ” means the number before the n-th term (recursive only) “a” or “a1 ” represents the first term (explicit only) “n” represents the term number “d” represents the common difference from one number to the next
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Arithmetic sequences:
Identifying the difference between recursive and explicit: Recursive Form: an = an-1 + d Explicit Form: an = a + (n – 1)d * a4 = ?; a3 = 6 ; d = * a4 = ?; a3 = 6 ; d = 2; a = 2; n =4 a4 = a a4 = 2 +(4 - 1) 2 a4 = a a4 = 2 +(3) 2 a4 = a4 = 2 +6 a4 = a4 = 8
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Arithmetic sequences:
Identify the following as being explicit or recursive: a5 = 10 +(5 - 1) 6 a7 = a a2 = 1 +(2 - 1) 12 a4 = a a3 = 17 +(3 - 1) 4 a6 = a a8 = 5 +(8 - 1) 5 a9 = a Answers on next slide.
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Arithmetic sequences:
Identify the following as being explicit or recursive: (ANSWERS) a5 = 10 +(5 - 1) 6 EXPLICIT a7 = a RECURSIVE a2 = 1 +(2 - 1) 12 EXPLICIT a4 = a RECURSIVE a6 = a a8 = 5 +(8 - 1) 5 a3 = 17 +(3 - 1) 4 a9 = a
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Arithmetic sequences:
Using the information, write the arithmetic sequence in the correct form. Recursive Form: an = an-1 + d Explicit Form: an = a + (n – 1)d 1. n = 5; d = 6; a = n = 5; d = 6 3. n = 7; d = -4 4. n = 7; d = -4; a = n = 5; d = 3; a = n = 2; d = 9 Answers on next slide.
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Arithmetic sequences:
Using the information, write the arithmetic sequence in the correct form and simplify. Recursive Form: an = an-1 + d Explicit Form: an = a + (n – 1)d (ANSWERS) 1. n = 5; d = 6; a = n = 5; d = 6 3. n = 7; d = -4 a5 = 2 + (5 – 1)6 a5 = a a7 = a a5 = 2 + (4)6 a5 = a a7 = a6 - 4 a5 = 26 *need a4 *need a6 4. n = 7; d = -4; a = n = 5; d = 3; a = n = 2; d = 9 Answers on next slide.
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Arithmetic sequences:
Using the given data set write the correct sequence recursively AND explicitly. Example: …. 1st 2nd 3rd 4th th th … Recursively an = an-1 + d Explicitly an = a + (n – 1)d d = 3 (the difference between the numbers) a = 2 (first term); d = 3 an = an an = 2 + (n – 1)3
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Arithmetic sequences: PRACTICE
Using the given data set write the correct sequence recursively AND explicitly. 1. 5 , 8 , 11 , 14 , 17 , ... 2. 26 , 31 , 36 , 41 , 46 , ... 3. 20 , 18 , 16 , 14 , 12 , ... 4. 45, 40, 35, 30, 25, … Answers on next slide.
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Arithmetic sequences: PRACTICE
Using the given data set write the correct sequence recursively AND explicitly. ANSWERS recursively explicitly 1. 5 , 8 , 11 , 14 , 17 , an = an an = 5 + (n – 1)3 2. 26 , 31 , 36 , 41 , 46 , ... 3. 20 , 18 , 16 , 14 , 12 , ... 4. 45, 40, 35, 30, 25, …
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Arithmetic sequencing application
Example one: Renting a backhoe costs a flat fee of $65 plus an additional $35 per hour. a. Write the first four terms of a sequence that represents the total cost of renting the backhoe for 1, 2, 3, and 4 hours. b. What is the common difference? C. What are the 5th, 24th, 48th and 72nd terms in the sequence? Answers on next slide.
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Geometric sequences Recursive Form: an = an-1 * r
Explicit Form: an = ar(n – 1) “an ” means the n-th term (the term you want) “an-1 ” means the number before the n-th term (recursive only) “a” or “a1 ” represents the first term (explicit only) “n” represents the term number “r” represents the common ratio
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geometric sequences: Identifying the difference between recursive and explicit: Recursive Form: an = an-1 * r Explicit Form: an = ar(n – 1) n = 5; a4 = 32; r = 2 n = 5; a = 4; r = 2 a5 = a5-1 * 2 a5 = 4*2(5 – 1) a5 = a4 * 2 a5 = 4*24 a5 = 32* 2 a5 = 4*24 a5 = a5 = 4*16 a5 = 64
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Geometric sequences: Identify the following as being explicit or recursive: a5 = 5*6(5 – 1) a3 = a3-1 *4 a4 = 8*9(4 – 1) a6 = a5-1 *12 a11 = 5*4(11 – 1) a32 = a32-1 *14 a93 = a93-1 *-2 a2 = 16*5(2 – 1) Answers on next slide.
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Geometric sequences: Identify the following as being explicit or recursive: (ANSWERS) a5 = 5*6(5 – 1) EXPLICIT a3 = a3-1 *4 RECURSIVE a4 = 8*9(4 – 1) EXPLICIT a6 = a5-1 *12 RECURSIVE a11 = 5*4(11 – 1) a32 = a32-1 *14 a93 = a93-1 *-2 a2 = 16*5(2 – 1)
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Geometric sequences: Using the information, write the arithmetic sequence in the correct form. Recursive Form: an = an-1 * r Explicit Form: an = ar(n – 1) 1. n = 5; r = 6; a = n = 5; r = 6 3. n = 7; r = -4 4. n = 7; r = -4; a = n = 5; r = 3; a = n = 2; r = 9 Answers on next slide.
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Geometric sequences: (ANSWERS)
Using the information, write the arithmetic sequence in the correct form. (ANSWERS) Recursive Form: an = an-1 * r Explicit Form: an = ar(n – 1) 1. n = 5; r = 6; a = n = 5; r = 6 3. n = 7; r = -4 a5 = 2*6(5 – 1) a5 = a5-1 *6 a7 = a7-1 *-4 a5 = 2*64 a5 = a4 *6 a7 = a6 *-4 a5 = 2* *need a4 *need a6 a5 = 2592 4. n = 7; r = -4; a = n = 5; r = 3; a = n = 2; r = 9
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Geometric sequences: Using the given data set write the correct sequence recursively AND explicitly. Example: …. 1st 2nd 3rd 4th th th … Recursive Form: an = an-1 * r Explicit Form: an = ar(n – 1) r = 2 (the ratio, divide 8/4 = 2, 16/2 = 2…) a = 4 (first term); r = 2 an = an-1 *2 an = 4*2(n – 1)
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Geometric sequences: practice
Using the given data set write the correct sequence recursively AND explicitly. 1. 1 , 2 , 4 , 8 , 16 , ... 2. 1 , 6 , 36 , 216 , 1296 , ... 3. 16 , -8 , 4 , -2 , 1 , ... , 108, -36, 12, -4… Answers on next slide.
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Geometric sequences: practice
Using the given data set write the correct sequence recursively AND explicitly. ANSWERS recursively explicitly 1. 1 , 2 , 4 , 8 , 16 , an = an-1 * an = 1*2(n – 1) 2. 1 , 6 , 36 , 216 , 1296 , an = 1*6(n – 1) 3. 16 , -8 , 4 , -2 , 1 , ... an = an-1 *(-1/2) , 108, -36, 12, -4…
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