Geometric and arithmetic sequences Recursive AND Explicit Forms

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.

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

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 = 2 * a4 = ?; a3 = 6 ; d = 2; a = 2; n =4 a4 = a4-1 + 2 a4 = 2 +(4 - 1) 2 a4 = a3 + 2 a4 = 2 +(3) 2 a4 = 6 + 2 a4 = 2 +6 a4 = 8 a4 = 8

Arithmetic sequences: Identify the following as being explicit or recursive: a5 = 10 +(5 - 1) 6 a7 = a7-1 + 16 a2 = 1 +(2 - 1) 12 a4 = a4-1 - 2 a3 = 17 +(3 - 1) 4 a6 = a6-1 - 9 a8 = 5 +(8 - 1) 5 a9 = a9-1 - 14 Answers on next slide.

Arithmetic sequences: Identify the following as being explicit or recursive: (ANSWERS) a5 = 10 +(5 - 1) 6 EXPLICIT a7 = a7-1 + 16 RECURSIVE a2 = 1 +(2 - 1) 12 EXPLICIT a4 = a4-1 - 2 RECURSIVE a6 = a6-1 - 9 a8 = 5 +(8 - 1) 5 a3 = 17 +(3 - 1) 4 a9 = a9-1 - 14

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 = 2 2. n = 5; d = 6 3. n = 7; d = -4 4. n = 7; d = -4; a =2 5. n = 5; d = 3; a = 7 6. n = 2; d = 9 Answers on next slide.

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 = 2 2. n = 5; d = 6 3. n = 7; d = -4 a5 = 2 + (5 – 1)6 a5 = a5-1 + 6 a7 = a7-1 + -4 a5 = 2 + (4)6 a5 = a4 + 6 a7 = a6 - 4 a5 = 12 *need a4 *need a6 4. n = 7; d = -4; a =2 5. n = 5; d = 3; a = 7 6. n = 2; d = 9 Answers on next slide.

Arithmetic sequences: Using the given data set write the correct sequence recursively AND explicitly. Example: 2 5 8 11 14 17…. 1st 2nd 3rd 4th 5th 6th … 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-1 + 3 an = 2 + (n – 1)3

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.

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-1 + 3 an = 5 + (n – 1)3 2. 26 , 31 , 36 , 41 , 46 , ... 3. 20 , 18 , 16 , 14 , 12 , ... 4. 45, 40, 35, 30, 25, …

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.

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

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 = 64 a5 = 4*16 a5 = 64

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.

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)

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 = 2 2. n = 5; r = 6 3. n = 7; r = -4 4. n = 7; r = -4; a =2 5. n = 5; r = 3; a = 7 6. n = 2; r = 9 Answers on next slide.

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 = 2 2. 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*1296 *need a4 *need a6 a5 = 2592 4. n = 7; r = -4; a =2 5. n = 5; r = 3; a = 7 6. n = 2; r = 9

Geometric sequences: Using the given data set write the correct sequence recursively AND explicitly. Example: 4 8 16 32 64 128…. 1st 2nd 3rd 4th 5th 6th … 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)

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 , ... 4. -324 , 108, -36, 12, -4… Answers on next slide.

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 *2 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) 4. -324 , 108, -36, 12, -4…