Recursive Sequence

From Recursive to Explicit Formula

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P.10 

Convert a recursive formula to an explicit formula

Multiple Choice

Given the recursive formula, find the explicit form
a_n = a_n-1 - 5
a₁ = 22

• a_n = 22 + (n-1)(-5)
• a_n = 22 + (n-1)(5)
• a_n = -5 + (n-1)(22)
• a_n = -16 + (n-1)(-5)

Multiple Choice

Given the recursive formula, find the explicit form
a_n = a_n-1 - 5
a₁ = 22

• a_n = 22 + (n-1)(-5)
• a_n = 22 + (n-1)(5)
• a_n = -5 + (n-1)(22)
• a_n = -16 + (n-1)(-5)

Multiple Choice

Write a recursive rule for the explicit rule of arithmetic sequence sequence

a_n = 17 -16(n-1)

• a₁ = 17
  a_n = a_n-1 + 16
• a₁ = 17
  a_n = a_n-1 - 17
• a₁ = 17
  a_n = a_n-1 - 15
• a₁ = 17
  a_n = a_n-1 - 16

Multiple Choice

Write a recursive formula for the sequence with explicit rule:

a_n=16 + (1/2)^n-1

• a₁=16

  a_n=(2)a_n-1

• a₁=16

  a_n=(1/2)a_n-1

• a₁=16

  a_n=a_n-1 + (1/2)

• a₁=16

  a_n=a_n-1 + (2)

Multiple Choice

Identify the rule as recursive or explicit, then find the first 4 terms of the sequence.
a₁ = 100; a_n = a_n-1 - 8

• Explicit; 100, 92, 84, 76
• Recursive; 100, 92, 84, 76
• Explicit; 100, 108, 116, 124
• Recursive; 100, 108, 116, 124

Multiple Choice

Find the first four terms of the sequence.

a_n+1= a_n + 9 where a₁= -2

• -2, -11, -20, -29

• 9, 7, 5, 3

• -2, 7, 16, 25

• 9, 18, 27, 36

Multiple Choice

Given the recursive formula, find the first four terms:
a_n = a_n-1 + 5
a₁ = -16

• -16, -21, -26, -31
• 5, -11, -27, -43
• -16, -80, -500, -200
• -16, -11, -6, -1

Multiple Choice

Find the first four terms of the sequence.

a_n= 2a_(n-1) + n where a₁= 0

• 0, 0, 0, 0

• 0, 1, 4, 12

• 0, 1, 2, 4

• 0, 2, 4, 8

Multiple Choice

Given the recursive formula, find the first four terms:
a_n = a_n-1 + 5
a₁ = -16

• -16, -21, -26, -31
• 5, -11, -27, -43
• -16, -80, -500, -200
• -16, -11, -6, -1

Multiple Choice

Given the recursive formula below, find the explicit formula.  
a₀ = 1
a_n = a_n-1 + 5

• a_n = 6 + 5n
• a_n = 1 + 5n

Multiple Choice

Determine the first 3 terms of the following sequence.

t₁= -3

t_n=t_n-1(3)

• -3, 9, 27

• -3, -9, -27

• -3, -6, -9

• -3, 6,9

Multiple Choice

Find the first five terms and common ratio using a recursive formula

a_n=a_n-1(2) a₁=2

• r=4

  32,16,8,4,2

• r=4

  2,4,8,16,32

• r=2

  32,16,8,4,2

• r=2

  2,4,8,16,32

Geometric Recursive

Multiple Choice

A sequence is represented below.

{–2, –6, –18, –54,  –162 ...}

 Which recursive notation could be used to represent the sequence?

• a₁ = -2

  a_n = -3 • a_{n - 1}

• a₁ = -2

  a_n = a_{n - 1} – 4

• a₁ = 2

  a_n = -3 • a_{n - 1}

• a₁ = -2

  a_n = 3 • a_{n - 1}

Multiple Choice

Given the first term and the common ratio of a geometric sequence find the explicit formula and the recursive formula.

$$a_1=3,\ r=-5$$  

Multiple Choice

Given the first term and the common ratio of a geometric sequence find the explicit formula and the recursive formula.

$$a_1=4,\ r=-5$$  

Multiple Choice

Given the explicit formula for a geometric sequence find the recursive formula.

$$a_n=-2\cdot6^{n-1}$$  

• $$a_n=a_{n-1}\cdot-6,\ a_1=-2$$  

• $$a_n=a_{n-1}\cdot-3,\ a_1=-3$$  

• $$a_n=a_{n-1}\cdot6,\ a_1=-2$$  

• $$a_n=a_{n-1}\cdot6,\ a_1=2$$  

Multiple Choice

Given the explicit formula for a geometric sequence find the recursive formula.

$$a_n=-3\cdot\left(\frac{1}{2}\right)^{n-1}$$  

• $$a_n=a_{n-1}\cdot-2,\ a_1=4$$  

• $$a_n=a_{n-1}\cdot-\frac{1}{2},\ a_1=4$$  

• $$a_n=a_{n-1}\cdot\frac{1}{2},\ a_1=-4$$  

• $$a_n=a_{n-1}\cdot\frac{1}{2},\ a_1=-3$$  

Multiple Choice

Given the recursive formula for a geometric sequence find the explicit formula. 

$$a_n=a_{n-1}\cdot-6,\ a_1=-4$$  

•   $$a_n=-4\cdot\left(4\right)^{n-1}$$  

•   $$a_n=-4\cdot\left(-6\right)^{n-1}$$  
  

• $$a_n=-12\cdot3^{n-1}$$  
  

• $$a_n=-4\cdot6^{n-1}$$  
  

Multiple Choice

Given the recursive formula for a geometric sequence find the explicit formula. 

$$a_n=a_{n-1}\cdot6,\ a_1=-3$$  

•   $$a_n=3\cdot\left(6\right)^{n-1}$$  

•   $$a_n=3\cdot\left(-6\right)^{n-1}$$  
  

• $$a_n=-3\cdot5^{n-1}$$  
  

• $$a_n=-3\cdot6^{n-1}$$