Worksheet: Explicit and Recursive Formulas of Geometric Sequences

In this worksheet, we will practice writing explicit and recursive formulas for geometric sequences and converting between the two types of formulas.

Q1:

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

  • A 7 6 × 1 2
  • B 7 6 × 2
  • C 7 6 × 2
  • D 7 6 × 1 2

Q2:

Find, in terms of 𝑛 , the general term of the sequence 2 , 4 , 8 , 1 6 , and find the order of the term whose value is 512.

  • A 4 , 𝑎
  • B 2 , 𝑎
  • C 4 , 𝑎
  • D 2 , 𝑎

Q3:

Find, in terms of 𝑛 , the general term of the arithmetic sequence 7 , 5 , 3 , 1 , .

  • A 𝑇 = 7 𝑛 + 2 𝑛
  • B 𝑇 = 7 𝑛 9 𝑛
  • C 𝑇 = 9 𝑛 + 2 𝑛
  • D 𝑇 = 2 𝑛 9 𝑛
  • E 𝑇 = 2 𝑛 𝑛

Q4:

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 1 )
  • C ( 1 )
  • D ( 1 )
  • E ( 1 )

Q5:

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 ( 𝑛 )
  • D ( 1 )
  • E ( 𝑛 )

Q6:

A geometric sequence starts with the term 𝑎 = 6 0 and has common ratio 3 . Find an explicit formula for 𝑎 𝑛 , 𝑛 0 .

  • A 𝑎 = ( 3 ) ( 6 ) 𝑛 𝑛
  • B 𝑎 = 2 ( 3 ) 𝑛 𝑛
  • C 𝑎 = ( 3 ) ( 6 ) 𝑛 𝑛 1
  • D 𝑎 = 2 ( 3 ) 𝑛 𝑛 + 1
  • E 𝑎 = 2 ( 3 ) 𝑛 𝑛 + 1

Q7:

The recursive formula for a geometric sequence is 𝑢 = 0 . 3 4 5 𝑢 and 𝑢 = 9 . 8 for 𝑛 1 . Give an explicit formula for the sequence.

  • A 𝑢 = 9 . 8 ( 0 . 3 4 5 ) for 𝑛 > 1
  • B 𝑢 = 9 . 8 6 . 4 1 9 ( 𝑛 1 ) for 𝑛 1
  • C 𝑢 = 3 . 3 8 𝑟 for 𝑛 1
  • D 𝑢 = 9 . 8 ( 0 . 3 4 5 ) for 𝑛 1
  • E 𝑢 = 9 . 8 ( 2 . 8 9 9 ) for 𝑛 1

Q8:

A sequence is defined by the recursive formula 𝑎 = 3 𝑎 2 , 𝑎 = 2 .

Find the first six terms of this sequence.

  • A 2 , 8 , 2 6 , 8 0 , 2 4 0 , 7 2 2
  • B 2 , 8 , 2 6 , 8 0 , 2 4 2 , 7 2 6
  • C 2 , 6 , 1 8 , 7 8 , 2 3 6 , 7 1 0
  • D 2 , 8 , 2 7 , 8 0 , 2 4 2 , 7 2 8
  • E 2 , 8 , 2 6 , 8 0 , 2 4 2 , 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 = 𝑎 1 .

  • A 𝑏 = ( 1 ) 3
  • B 𝑏 = 3
  • C 𝑏 = ( 1 ) 3
  • D 𝑏 = 3 𝑛
  • E 𝑏 = ( 3 )

Using your answer to the previous part, find an explicit formula for 𝑎 .

  • A 𝑎 = 1 3
  • B 𝑎 = 1 3
  • C 𝑎 = ( 3 ) + 1
  • D 𝑎 = 1 + 3
  • E 𝑎 = 3 𝑛 + 1

Consider the sequence defined by the recursive formula 𝑐 = 7 𝑐 + 4 𝑐 = 2 , . Find the value of 𝑘 for which 𝑑 = 𝑐 + 𝑘 is a geometric sequence with common ratio 7.

  • A 𝑘 = 3 2
  • B 𝑘 = 4 7
  • C 𝑘 = 4
  • D 𝑘 = 2 3
  • E 𝑘 = 4

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

  • A 𝑐 = 7 5
  • B 𝑐 = 7 2 3
  • C 𝑐 = 4 3 7 + 2 3
  • D 𝑐 = 8 3 7 2 3
  • E 𝑐 = 7 4

Q9:

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

Q10:

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

  • A 1 9 ( )
  • B 1 9
  • C 1 9 1
  • D 1 9 ( )

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 , 4 𝜋 , 6 𝜋 , 8 𝜋 , .

  • A c o s ( 2 ( 𝑛 + 1 ) 𝜋 )
  • B c o s ( 4 𝑛 𝜋 )
  • C c o s ( 2 ( 𝑛 1 ) 𝜋 )
  • D c o s ( 2 𝑛 𝜋 )

Q12:

Find, in terms of 𝑛 , the general term of the sequence ( 1 8 , 7 2 , 1 6 2 , 2 8 8 , ) .

  • A 1 8 𝑛 3
  • B 1 8 𝑛
  • C 1 9 𝑛 1
  • D 1 8 𝑛 2
  • E 1 7 𝑛 + 1

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