1. Geometric and arithmetic sequences
Recursive AND Explicit Forms

2. 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.

3. 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

4. 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

5. 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.

6. 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

7. 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.

8. 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 = 12 *need a4 *need a6
4. n = 7; d = -4; a = n = 5; d = 3; a = n = 2; d = 9
Answers on next slide.

9. 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

10. 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.

11. 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, …

12. 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.

13. 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

14. 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

15. 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.

16. 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)

17. 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.

18. 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

19. 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)

20. 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.

21. 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…