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Worksheet: Recursive Sequences

Q1:

The sequence π‘Ž  , where 𝑛 β‰₯ 1 , is given by

List the next 6 terms π‘Ž , … , π‘Ž     .

  • A π‘Ž = βˆ’ 6   , π‘Ž = 6   , π‘Ž = βˆ’ 7   , π‘Ž = 7  οŠͺ , π‘Ž = 7   , π‘Ž = 8  
  • B π‘Ž = 5   , π‘Ž = 6   , π‘Ž = βˆ’ 6   , π‘Ž = βˆ’ 7  οŠͺ , π‘Ž = 7   , π‘Ž = 8  
  • C π‘Ž = 6   , π‘Ž = βˆ’ 6   , π‘Ž = βˆ’ 6   , π‘Ž = 7  οŠͺ , π‘Ž = 7   , π‘Ž = βˆ’ 8  
  • D π‘Ž = βˆ’ 5   , π‘Ž = βˆ’ 6   , π‘Ž = 6   , π‘Ž = 7  οŠͺ , π‘Ž = βˆ’ 7   , π‘Ž = βˆ’ 8  
  • E π‘Ž = βˆ’ 5   , π‘Ž = 6   , π‘Ž = 6   , π‘Ž = βˆ’ 7  οŠͺ , π‘Ž = βˆ’ 7   , π‘Ž = 8  

By listing the elements π‘Ž , π‘Ž , π‘Ž , π‘Ž , …      , give a formula for π‘Ž οŠͺ    , in terms of 𝑛 , for 𝑛 β‰₯ 1 .

  • A π‘Ž = 2 ( 𝑛 βˆ’ 1 ) οŠͺ   
  • B π‘Ž = ( 𝑛 + 1 ) οŠͺ   
  • C π‘Ž = 2 ( 𝑛 + 1 ) οŠͺ   
  • D π‘Ž = ( 𝑛 βˆ’ 1 ) οŠͺ   
  • E π‘Ž = ( 2 𝑛 βˆ’ 1 ) οŠͺ   

Give a formula for π‘Ž οŠͺ    , in terms of 𝑛 , for 𝑛 β‰₯ 1 .

  • A π‘Ž = 𝑛 + 1 οŠͺ   
  • B π‘Ž = 2 𝑛 + 1 οŠͺ   
  • C π‘Ž = 2 𝑛 βˆ’ 1 οŠͺ   
  • D π‘Ž = 2 ( 𝑛 βˆ’ 1 ) οŠͺ   
  • E π‘Ž = 2 ( 𝑛 + 1 ) οŠͺ   

Give a formula for π‘Ž οŠͺ    , in terms of 𝑛 , for 𝑛 β‰₯ 1 .

  • A π‘Ž = 1 βˆ’ 2 𝑛 οŠͺ   
  • B π‘Ž = 2 + 𝑛 οŠͺ   
  • C π‘Ž = 2 βˆ’ 𝑛 οŠͺ   
  • D π‘Ž = 1 + 2 𝑛 οŠͺ   
  • E π‘Ž = 1 βˆ’ 𝑛 οŠͺ   

Give a formula for π‘Ž οŠͺ  , in terms of 𝑛 , for 𝑛 β‰₯ 1 .

  • A π‘Ž = 1 + 2 𝑛 οŠͺ 
  • B π‘Ž = 2 𝑛 οŠͺ 
  • C π‘Ž = 1 βˆ’ 2 𝑛 οŠͺ 
  • D π‘Ž = βˆ’ 2 𝑛 οŠͺ 
  • E π‘Ž = 2 βˆ’ 𝑛 οŠͺ 

What is π‘Ž    οŠͺ  ?

  • A π‘Ž = 6 1 7 2    οŠͺ 
  • B π‘Ž = 6 7 1 0    οŠͺ 
  • C π‘Ž = βˆ’ 6 1 7 0    οŠͺ 
  • D π‘Ž = 6 1 7 0    οŠͺ 
  • E π‘Ž = βˆ’ 6 1 7 2    οŠͺ 

Solve π‘Ž = 1 7  for 𝑛 .

  • A 𝑛 = 3 2
  • B 𝑛 = 3 4
  • C 𝑛 = 3 5
  • D 𝑛 = 3 7
  • E 𝑛 = 3 3

What is the range of the function π‘Ž  ?

  • Athe set of negative integers
  • Bthe set of negative rationals
  • Cthe set of postive rationals
  • Dthe set of all integers
  • Ethe set of positive integers

Q2:

Find the first five terms of the sequence with general term , where and .

  • A
  • B
  • C
  • D

Q3:

Find the first five terms of the sequence with general term , where and .

  • A
  • B
  • C
  • D

Q4:

Find the first five terms of the sequence with general term , where and .

  • A
  • B
  • C
  • D

Q5:

Find the first five terms of the sequence with general term , where and .

  • A
  • B
  • C
  • D

Q6:

Find the first five terms of the sequence with general term , where and .

  • A
  • B
  • C
  • D

Q7:

The term in a sequence is given by . Find the first six terms of this sequence, given that and .

  • A
  • B
  • C
  • D

Q8:

The term in a sequence is given by . Find the first six terms of this sequence, given that and .

  • A
  • B
  • C
  • D

Q9:

The term in a sequence is given by . Find the first six terms of this sequence, given that and .

  • A
  • B
  • C
  • D

Q10:

The term in a sequence is given by . Find the first six terms of this sequence, given that .

  • A
  • B
  • C
  • D

Q11:

Find given and .

  • A
  • B
  • C
  • D

Q12:

Given that π‘Ž = 8 1 and that π‘Ž = 1 2 π‘Ž 𝑛 + 1 𝑛 for 𝑛 β‰₯ 1 , find a formula for π‘Ž 𝑛 in terms of 𝑛 .

  • A π‘Ž = ο€Ό 1 2  𝑛 𝑛 + 1
  • B π‘Ž = 8 ο€Ή 2  𝑛 𝑛 βˆ’ 4
  • C π‘Ž = 2 𝑛 𝑛 βˆ’ 1
  • D π‘Ž = 2 𝑛 4 βˆ’ 𝑛
  • E π‘Ž = 2 𝑛 8 βˆ’ 𝑛

Q13:

Find the arithmetic sequence in which and .

  • A
  • B
  • C
  • D

Q14:

Find the arithmetic sequence in which and .

  • A
  • B
  • C
  • D

Q15:

Consider the following sequence of dots.

What is the function 𝑓 such that 𝑓 ( 𝑛 ) is the number of dots in the 𝑛 th pattern?

  • A 𝑓 ( 𝑛 ) = 𝑛 ( 2 𝑛 + 1 )
  • B 𝑓 ( 𝑛 ) = ( 2 𝑛 + 1 )
  • C 𝑓 ( 𝑛 ) = 2 ( 𝑛 βˆ’ 1 )
  • D 𝑓 ( 𝑛 ) = 𝑛 ( 2 𝑛 βˆ’ 1 ) = 2 𝑛 βˆ’ 𝑛 2
  • E 𝑓 ( 𝑛 ) = 2 ( 𝑛 + 1 )

Q16:

The graph represents the triangle wave function 𝑇 ( π‘₯ ) , which is periodic, piecewise linear, and defined for all real numbers.

List the values of 𝑇 ( 0 ) , 𝑇 ( βˆ’ 1 ) , and 𝑇 ( 1 2 3 4 ) .

  • A1, 1, 1
  • B0, 1, 1
  • C0, βˆ’ 1 , 1
  • D0, 0, 0
  • E1, βˆ’ 1 , 0

List the values of 𝑇 ο€Ό 1 2  , 𝑇 ο€Ό 3 2  , 𝑇 ο€Ό 5 2  , and 𝑇 ο€Ό 1 2 3 3 2  .

  • A1, βˆ’ 1 , 1, 1
  • B βˆ’ 1 , βˆ’ 1 , 1, 1
  • C1, 1, 1, 1
  • D1, βˆ’ 1 , 1, βˆ’ 1
  • E1, 1, βˆ’ 1 , 1

What is 𝑇 ο€Ό βˆ’ 4 9 3 3 2  ?

  • A βˆ’ 1
  • B1
  • C0
  • Dundefined

If we are given that 𝑇 ( 𝑏 ) is negative, what can we conclude about the number 𝑏 ?

  • A There is some integer 𝑛 for which 2 𝑛 + 1 < 𝑏 < 2 𝑛 + 2 .
  • B 𝑏 is an even integer.
  • C There is some integer 𝑛 for which 2 𝑛 < 𝑏 < 2 𝑛 + 1 .
  • D 𝑏 is an odd integer.

Find the equation of the line segment on which the point ( πœ‹ , 𝑇 ( πœ‹ ) ) lies.

  • A 𝑦 = βˆ’ 1 2 ( π‘₯ βˆ’ 1 2 )
  • B 𝑦 = 2 ( 3 π‘₯ βˆ’ 1 )
  • C 𝑦 = βˆ’ 2 ( π‘₯ + 3 )
  • D 𝑦 = βˆ’ 2 ( π‘₯ βˆ’ 3 )
  • E 𝑦 = βˆ’ 4 ( π‘₯ βˆ’ 3 )

Hence find the value of 𝑇 ( πœ‹ ) correct to 3 decimal places.

  • A16.850
  • B βˆ’ 0 . 5 6 6
  • C12.283
  • D βˆ’ 0 . 2 8 3
  • E4.429

Q17:

Find, in terms of the general term of the sequence which satisfies the relation , where and .

  • A
  • B
  • C
  • D