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Lesson: Recursive Sequences

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17:53

Sample Question Videos

Worksheet • 17 Questions • 1 Video

Q1:

Find the first five terms of the sequence with general term 𝑇 = 𝑇 + 5 𝑛 + 1 𝑛 , where 𝑛 β‰₯ 1 and 𝑇 = βˆ’ 1 3 1 .

  • A ( βˆ’ 1 3 , βˆ’ 8 , βˆ’ 3 , 2 , 7 )
  • B ( βˆ’ 1 3 , βˆ’ 1 8 , βˆ’ 2 3 , βˆ’ 2 8 , βˆ’ 3 3 )
  • C ( βˆ’ 1 8 , βˆ’ 2 3 , βˆ’ 2 8 , βˆ’ 3 3 , βˆ’ 3 8 )
  • D ( βˆ’ 8 , βˆ’ 3 , 2 , 7 , 1 2 )

Q2:

The 𝑛 t h term in a sequence is given by 𝑇 = 𝑇 + 𝑇 𝑛 + 2 𝑛 + 1 𝑛 . Find the first six terms of this sequence, given that 𝑇 = 0 1 and 𝑇 = 1 2 .

  • A ( 0 , 1 , 1 , 2 , 3 , 5 )
  • B ( 1 , 2 , 3 , 5 , 8 , 1 3 )
  • C ( 0 , 1 , 1 , 2 , 3 , 4 )
  • D ( 0 , 1 , 2 , 3 , 5 , 8 )

Q3:

Find the arithmetic sequence in which 𝑇 = βˆ’ 1 0 0 1 and 𝑇 = 4 𝑇 4 𝑛 𝑛 .

  • A ( βˆ’ 1 0 0 , βˆ’ 2 0 0 , βˆ’ 3 0 0 , β‹― )
  • B ( βˆ’ 1 0 0 , βˆ’ 3 0 0 , βˆ’ 5 0 0 , β‹― )
  • C ( βˆ’ 1 0 0 , βˆ’ 4 0 0 , βˆ’ 5 0 0 , β‹― )
  • D ( βˆ’ 1 0 0 , βˆ’ 3 0 0 , βˆ’ 4 0 0 , β‹― )

Q4:

Given that and that for , find a formula for in terms of .

  • A
  • B
  • C
  • D
  • E

Q5:

Find 𝑇 + 𝑇 + 𝑇 1 3 1 4 1 5 given 𝑇 = βˆ’ 3 1 and 𝑇 = 𝑇 + 5 8 𝑛 𝑛 βˆ’ 1 .

  • A 1 2 3 8
  • B 6 9 4
  • C βˆ’ 9 2 1 8
  • D βˆ’ 2 6 7 8

Q6:

The 𝑛 t h term in a sequence is given by 𝑇 = 𝑛 𝑇 𝑛 + 1 𝑛 . Find the first six terms of this sequence, given that 𝑇 = βˆ’ 1 1 8 1 .

  • A ( βˆ’ 1 1 8 , βˆ’ 1 1 8 , βˆ’ 2 3 6 , βˆ’ 7 0 8 , βˆ’ 2 8 3 2 , βˆ’ 1 4 1 6 0 )
  • B ( βˆ’ 1 1 8 , βˆ’ 2 3 6 , βˆ’ 3 5 4 , βˆ’ 4 7 2 , βˆ’ 5 9 0 , βˆ’ 7 0 8 )
  • C ( βˆ’ 2 3 6 , βˆ’ 3 5 4 , βˆ’ 4 7 2 , βˆ’ 5 9 0 , βˆ’ 7 0 8 , βˆ’ 8 2 6 )
  • D ( βˆ’ 1 1 8 , βˆ’ 2 3 6 , βˆ’ 7 0 8 , βˆ’ 2 8 3 2 , βˆ’ 1 4 1 6 0 , βˆ’ 8 4 9 6 0 )

Q7:

Find, in terms of 𝑛 the general term of the sequence which satisfies the relation 𝑇 = 2 2 𝑇 𝑛 + 1 𝑛 , where 𝑛 β‰₯ 1 and 𝑇 = 2 2 1 .

  • A ( 2 2 ) 𝑛
  • B 2 2 𝑛
  • C βˆ’ 2 2 𝑛
  • D ( βˆ’ 2 2 ) 𝑛

Q8:

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

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

  • A π‘Ž = βˆ’ 5   , π‘Ž = βˆ’ 6   , π‘Ž = 6   , π‘Ž = 7  οŠͺ , π‘Ž = βˆ’ 7   , π‘Ž = βˆ’ 8  
  • B π‘Ž = βˆ’ 5   , π‘Ž = 6   , π‘Ž = 6   , π‘Ž = βˆ’ 7  οŠͺ , π‘Ž = βˆ’ 7   , π‘Ž = 8  
  • C π‘Ž = 5   , π‘Ž = 6   , π‘Ž = βˆ’ 6   , π‘Ž = βˆ’ 7  οŠͺ , π‘Ž = 7   , π‘Ž = 8  
  • D π‘Ž = 6   , π‘Ž = βˆ’ 6   , π‘Ž = βˆ’ 6   , π‘Ž = 7  οŠͺ , π‘Ž = 7   , π‘Ž = βˆ’ 8  
  • E π‘Ž = βˆ’ 6   , π‘Ž = 6   , π‘Ž = βˆ’ 7   , π‘Ž = 7  οŠͺ , π‘Ž = 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 = ( 2 𝑛 βˆ’ 1 ) οŠͺ   
  • C π‘Ž = ( 𝑛 βˆ’ 1 ) οŠͺ   
  • D π‘Ž = ( 𝑛 + 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 = 2 𝑛 βˆ’ 1 οŠͺ   
  • B π‘Ž = 𝑛 + 1 οŠͺ   
  • C π‘Ž = 2 𝑛 + 1 οŠͺ   
  • D π‘Ž = 2 ( 𝑛 + 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 = 1 βˆ’ 2 𝑛 οŠͺ   
  • B π‘Ž = 2 + 𝑛 οŠͺ   
  • C π‘Ž = 1 + 2 𝑛 οŠͺ   
  • D π‘Ž = 2 βˆ’ 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 = βˆ’ 2 𝑛 οŠͺ 
  • B π‘Ž = 2 βˆ’ 𝑛 οŠͺ 
  • C π‘Ž = 1 βˆ’ 2 𝑛 οŠͺ 
  • D π‘Ž = 1 + 2 𝑛 οŠͺ 
  • E π‘Ž = 2 𝑛 οŠͺ 

What is π‘Ž    οŠͺ  ?

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

Solve π‘Ž = 1 7  for 𝑛 .

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

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

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

Q9:

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

Q10:

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 ) .

  • A0, 0, 0
  • B1, βˆ’ 1 , 0
  • C0, 1, 1
  • D0, βˆ’ 1 , 1
  • E1, 1, 1

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
  • B1, 1, βˆ’ 1 , 1
  • C1, βˆ’ 1 , 1, βˆ’ 1
  • D βˆ’ 1 , βˆ’ 1 , 1, 1
  • E1, 1, 1, 1

What is 𝑇 ο€Ό βˆ’ 4 9 3 3 2  ?

  • A βˆ’ 1
  • B1
  • Cundefined
  • D0

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 There is some integer 𝑛 for which 2 𝑛 < 𝑏 < 2 𝑛 + 1 .
  • C 𝑏 is an even integer.
  • 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’ 2 ( π‘₯ βˆ’ 3 )
  • B 𝑦 = βˆ’ 4 ( π‘₯ βˆ’ 3 )
  • C 𝑦 = βˆ’ 2 ( π‘₯ + 3 )
  • D 𝑦 = βˆ’ 1 2 ( π‘₯ βˆ’ 1 2 )
  • E 𝑦 = 2 ( 3 π‘₯ βˆ’ 1 )

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

  • A βˆ’ 0 . 2 8 3
  • B βˆ’ 0 . 5 6 6
  • C12.283
  • D4.429
  • E16.850

Q11:

Given the sequence defined by 𝑇 = 𝑇 + 𝑛 π‘₯ 𝑛 + 1 𝑛 , where 𝑇 = 2 7 1 and 𝑇 = βˆ’ 7 8 3 , find the value of π‘₯ .

Q12:

Find the first five terms of the sequence with general term 𝑇 = 𝑇 βˆ’ 1 3 𝑛 + 1 𝑛 , where 𝑛 β‰₯ 1 and 𝑇 = 1 1 .

  • A ( 1 , βˆ’ 1 2 , βˆ’ 2 5 , βˆ’ 3 8 , βˆ’ 5 1 )
  • B ( 1 , 1 4 , 2 7 , 4 0 , 5 3 )
  • C ( 1 4 , 2 7 , 4 0 , 5 3 , 6 6 )
  • D ( βˆ’ 1 2 , βˆ’ 2 5 , βˆ’ 3 8 , βˆ’ 5 1 , βˆ’ 6 4 )

Q13:

Find the first five terms of the sequence with general term 𝑇 = 𝑇 βˆ’ 3 𝑛 + 1 𝑛 , where 𝑛 β‰₯ 1 and 𝑇 = 1 9 1 .

  • A ( 1 9 , 1 6 , 1 3 , 1 0 , 7 )
  • B ( 1 9 , 2 2 , 2 5 , 2 8 , 3 1 )
  • C ( 2 2 , 2 5 , 2 8 , 3 1 , 3 4 )
  • D ( 1 6 , 1 3 , 1 0 , 7 , 4 )

Q14:

Find the first five terms of the sequence with general term 𝑇 = 𝑇 + 2 6 𝑛 + 1 𝑛 , where 𝑛 β‰₯ 1 and 𝑇 = 2 0 1 .

  • A ( 2 0 , 4 6 , 7 2 , 9 8 , 1 2 4 )
  • B ( 2 0 , βˆ’ 6 , βˆ’ 3 2 , βˆ’ 5 8 , βˆ’ 8 4 )
  • C ( βˆ’ 6 , βˆ’ 3 2 , βˆ’ 5 8 , βˆ’ 8 4 , βˆ’ 1 1 0 )
  • D ( 4 6 , 7 2 , 9 8 , 1 2 4 , 1 5 0 )

Q15:

Find the first five terms of the sequence with general term 𝑇 = 𝑇 + 3 𝑛 + 1 𝑛 , where 𝑛 β‰₯ 1 and 𝑇 = βˆ’ 4 1 .

  • A ( βˆ’ 4 , βˆ’ 1 , 2 , 5 , 8 )
  • B ( βˆ’ 4 , βˆ’ 7 , βˆ’ 1 0 , βˆ’ 1 3 , βˆ’ 1 6 )
  • C ( βˆ’ 7 , βˆ’ 1 0 , βˆ’ 1 3 , βˆ’ 1 6 , βˆ’ 1 9 )
  • D ( βˆ’ 1 , 2 , 5 , 8 , 1 1 )

Q16:

The 𝑛 t h term in a sequence is given by 𝑇 = 𝑇 + 𝑇 𝑛 + 2 𝑛 + 1 𝑛 . Find the first six terms of this sequence, given that 𝑇 = 9 1 and 𝑇 = 1 1 1 2 .

  • A ( 9 , 1 1 1 , 1 2 0 , 2 3 1 , 3 5 1 , 5 8 2 )
  • B ( 1 2 0 , 2 3 1 , 3 5 1 , 5 8 2 , 9 3 3 , 1 5 1 5 )
  • C ( 9 , 1 1 1 , 1 2 0 , 2 3 1 , 3 5 1 , 4 7 1 )
  • D ( 9 , 1 2 0 , 2 3 1 , 3 5 1 , 5 8 2 , 9 3 3 )

Q17:

The 𝑛 t h term in a sequence is given by 𝑇 = 𝑇 + 𝑇 𝑛 + 2 𝑛 + 1 𝑛 . Find the first six terms of this sequence, given that 𝑇 = 1 6 3 1 and 𝑇 = βˆ’ 1 3 0 2 .

  • A ( 1 6 3 , βˆ’ 1 3 0 , 3 3 , βˆ’ 9 7 , βˆ’ 6 4 , βˆ’ 1 6 1 )
  • B ( 3 3 , βˆ’ 9 7 , βˆ’ 6 4 , βˆ’ 1 6 1 , βˆ’ 2 2 5 , βˆ’ 3 8 6 )
  • C ( 1 6 3 , βˆ’ 1 3 0 , 3 3 , βˆ’ 9 7 , βˆ’ 6 4 , βˆ’ 3 1 )
  • D ( 1 6 3 , 3 3 , βˆ’ 9 7 , βˆ’ 6 4 , βˆ’ 1 6 1 , βˆ’ 2 2 5 )
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