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Worksheet: Explicit and Recursive Formulas of Geometric Sequences

Q1:

Find, in terms of 𝑛 , the general term of the geometric sequence βˆ’ 7 6 , βˆ’ 3 8 , βˆ’ 1 9 , βˆ’ 1 9 2 , … .

  • A βˆ’ 7 6 Γ— ο€Ό 1 2  𝑛
  • B βˆ’ 7 6 Γ— 2 𝑛 βˆ’ 1
  • C βˆ’ 7 6 Γ— 2 𝑛
  • D βˆ’ 7 6 Γ— ο€Ό 1 2  𝑛 βˆ’ 1

Q2:

Find, in terms of 𝑛 , the general term of the geometric sequence βˆ’ 8 6 , βˆ’ 4 3 , βˆ’ 4 3 2 , βˆ’ 4 3 4 , … .

  • A βˆ’ 8 6 Γ— ο€Ό 1 2  𝑛
  • B βˆ’ 8 6 Γ— 2 𝑛 βˆ’ 1
  • C βˆ’ 8 6 Γ— 2 𝑛
  • D βˆ’ 8 6 Γ— ο€Ό 1 2  𝑛 βˆ’ 1

Q3:

Find, in terms of 𝑛 , the general term of the geometric sequence βˆ’ 4 8 , βˆ’ 1 2 , βˆ’ 3 , βˆ’ 3 4 , … .

  • A βˆ’ 4 8 Γ— ο€Ό 1 4  𝑛
  • B βˆ’ 4 8 Γ— 4 𝑛 βˆ’ 1
  • C βˆ’ 4 8 Γ— 4 𝑛
  • D βˆ’ 4 8 Γ— ο€Ό 1 4  𝑛 βˆ’ 1

Q4:

A geometric sequence starts with the term π‘Ž = βˆ’ 6 0 and has common ratio βˆ’ 3 . Find an explicit formula for π‘Ž 𝑛 , 𝑛 β‰₯ 0 .

  • A π‘Ž = ( βˆ’ 3 ) ( βˆ’ 6 ) 𝑛 𝑛
  • B π‘Ž = 2 ( βˆ’ 3 ) 𝑛 𝑛
  • C π‘Ž = ( βˆ’ 3 ) ( βˆ’ 6 ) 𝑛 𝑛 βˆ’ 1
  • D π‘Ž = 2 ( βˆ’ 3 ) 𝑛 𝑛 + 1
  • E π‘Ž = βˆ’ 2 ( βˆ’ 3 ) 𝑛 𝑛 + 1

Q5:

The recursive formula for a geometric sequence is 𝑒 = 0 . 3 4 5 𝑒     and 𝑒 = 9 . 8  for 𝑛 β‰₯ 1 . Give an explicit formula for the sequence.

  • A 𝑒 = 9 . 8 ( 0 . 3 4 5 )   for 𝑛 > 1
  • B 𝑒 = 9 . 8 βˆ’ 6 . 4 1 9 ( 𝑛 βˆ’ 1 )  for 𝑛 β‰₯ 1
  • C 𝑒 = 3 . 3 8 π‘Ÿ     for 𝑛 β‰₯ 1
  • D 𝑒 = 9 . 8 ( 0 . 3 4 5 )     for 𝑛 β‰₯ 1
  • E 𝑒 = 9 . 8 ( 2 . 8 9 9 )     for 𝑛 β‰₯ 1

Q6:

Using n to represent the position of a term of this sequence, and starting at 𝑛 = 1 , write an expression to describe the following sequence.

  • A ( 1 )  
  • B ( 𝑛 )  
  • C ( βˆ’ 𝑛 )  
  • D ( βˆ’ 1 ) 
  • E ( 𝑛 ) 

Q7:

Find, in terms of 𝑛 , the general term of the sequence ο€Ό 1 , 1 9 , 1 8 1 , 1 7 2 9  .

  • A 1 9 ( 𝑛 + 1 )
  • B 1 9 𝑛
  • C 1 9 βˆ’ 1 𝑛
  • D 1 9 ( 𝑛 βˆ’ 1 )

Q8:

The count of a termite population from week to week is modeled by the recursion 𝑝 = 1 . 0 5 𝑝     . What recursion would be used if the population was counted monthly? Use the average number of weeks per month over one year knowing there are 52 weeks or 12 months in a year, and approximate your answer to two decimal places.

  • A 𝑝 = 4 . 2 0 𝑝    
  • B 𝑝 = 1 . 2 6 𝑝    
  • C 𝑝 = 2 . 5 4 𝑝    
  • D 𝑝 = 1 . 2 4 𝑝    
  • E 𝑝 = 0 . 2 6 𝑝    

Q9:

A sequence is defined by the recursive formula π‘Ž = 3 π‘Ž βˆ’ 2 , π‘Ž = βˆ’ 2      .

Find the first six terms of this sequence.

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

Is this sequence arithmetic, geometric, both, or neither?

  • ABoth geometric and arithmetic
  • BArithmetic
  • CGeometric
  • DNeither geometric nor arithmetic

Find an explicit formula for 𝑏  , where 𝑏 = π‘Ž βˆ’ 1   .

  • A 𝑏 = ( βˆ’ 1 ) 3    
  • B 𝑏 = 3  
  • C 𝑏 = ( βˆ’ 1 ) 3  
  • D 𝑏 = βˆ’ 3 𝑛 
  • E 𝑏 = ( βˆ’ 3 )  

Using your answer to the previous part, find an explicit formula for π‘Ž  .

  • A π‘Ž = 1 βˆ’ 3  
  • B π‘Ž = 1 βˆ’ 3    
  • C π‘Ž = ( βˆ’ 3 ) + 1  
  • D π‘Ž = 1 + 3  
  • E π‘Ž = βˆ’ 3 𝑛 + 1 

Consider the sequence defined by the recursive formula 𝑐 = 7 𝑐 + 4 𝑐 = 2      , . Find the value of π‘˜ for which 𝑑 = 𝑐 + π‘˜   is a geometric sequence with common ratio 7.

  • A π‘˜ = 3 2
  • B π‘˜ = βˆ’ 4 7
  • C π‘˜ = βˆ’ 4
  • D π‘˜ = 2 3
  • E π‘˜ = 4

Using your answer to the previous part, derive an explicit formula for 𝑐  .

  • A 𝑐 = 7 βˆ’ 5  
  • B 𝑐 = 7 βˆ’ 2 3  
  • C 𝑐 = 4 3 7 + 2 3    
  • D 𝑐 = 8 3 7 βˆ’ 2 3    
  • E 𝑐 = 7 βˆ’ 4  

Q10:

Using 𝑛 to represent the position of a term of this sequence, and starting at 𝑛 = 1 , write an expression to describe the sequence

  • A ( 𝑛 + 1 ) 
  • B ( 𝑛 βˆ’ 1 )  
  • C ( βˆ’ 1 ) 
  • D ( βˆ’ 1 )   
  • E ( 1 )   

Q11:

Find, in terms of 𝑛 , the general term of the sequence c o s c o s c o s c o s 2 πœ‹ , 4 πœ‹ , 6 πœ‹ , 8 πœ‹ , … .

  • A c o s ( 2 ( 𝑛 + 1 ) πœ‹ )
  • B c o s ( 4 𝑛 πœ‹ )
  • C c o s ( 2 ( 𝑛 βˆ’ 1 ) πœ‹ )
  • D c o s ( 2 𝑛 πœ‹ )

Q12:

Find, in terms of , the general term of the sequence and find the order of the term whose value is 512.

  • A ,
  • B ,
  • C ,
  • D ,