Worksheet: Introduction to Sequences

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In this worksheet, we will practice identifying, from the pattern of a given sequence, whether it is arithmetic, geometric, or neither and how to differentiate between finite and infinite sequences.

Q1:

Find the range of the infinite arithmetic sequence represented in the figure below.

  • A { 4 , 0 , − 4 , − 8 , … }
  • B { 1 , 2 , 3 , 4 , … }
  • C { 4 , 0 , − 4 , − 8 }
  • D ℝ
  • E [ − 8 , 4 ]

Q2:

Is the sequence (−9,−6,−3,0,…,21) finite or infinite?

  • Ainfinite
  • Bfinite

Q3:

Is the sequence with general term 3𝑛+79, where 𝑛∈ℤ, finite or infinite?

  • Afinite
  • Binfinite

Q4:

Is each function whose domain is ℤ a sequence?

  • Ayes
  • Bno

Q5:

If we consider a sequence to be a function, what is the function’s domain?

  • A ℚ
  • B ℤ
  • C ℝ
  • D ℤ
  • E ℝ

Q6:

List the first 10 elements of the sequence where the 𝑛th term is defined as the remainder when 𝑛 is divided by 4.

  • A 1 , 2 , 3 , 1 , 2 , 3 , 1 , 2 , 3 , 1 ,
  • B 2 , 3 , 0 , 2 , 3 , 0 , 2 , 3 , 0 , 2 ,
  • C 1 , 2 , 0 , 1 , 2 , 0 , 1 , 2 , 0 , 1 ,
  • D 1 , 2 , 3 , 0 , 1 , 2 , 3 , 0 , 1 , 2 ,
  • E 1 , 3 , 0 , 1 , 3 , 0 , 1 , 3 , 0 , 1 ,

What is the range of this sequence?

  • A { 0 , 1 , 2 , 3 }
  • B { 0 , 1 , 2 }
  • C { 0 , 1 , 3 }
  • D { 1 , 2 , 3 }
  • E { 0 , 2 , 3 }

Q7:

Consider the infinite sequence 4,7,10,13,16,…. We can think of this sequence as a function whose graph is partially sketched.

Describe the domain of the function.

  • AAll positive integers: 1,2,3,4,5,…
  • B { 0 , 1 , 2 , 3 , 4 , 5 , … }
  • C { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 }
  • DAll integers
  • E ( 1 , ∞ )

Describe the range of the function.

  • AAll numbers 𝑦 satisfying 𝑦=3𝑥+1 for reals 𝑥≥1
  • BAll numbers 𝑦 satisfying 𝑦=3𝑥+1 for integers 𝑥≥0
  • CAll numbers 𝑦 satisfying 𝑦=3𝑥+1 for integers 1≤𝑥≤8
  • DAll numbers 𝑦 satisfying 𝑦=3𝑥+1 for integers 𝑥≥1
  • EAll numbers 𝑦 satisfying 𝑦=3𝑥+1 for integers −∞<𝑥<∞

Q8:

The graph of the first six terms of an arithmetic sequence is shown.

Write, in the form 𝑦=𝑚𝑥+𝑏, an equation for the sequence.

  • A 𝑦 = − 5 𝑥 + 8
  • B 𝑦 = − 2 𝑥 + 5
  • C 𝑦 = − 2 𝑥 − 5
  • D 𝑦 = 2 𝑥 − 5
  • E 𝑦 = 2 𝑥 + 5

Find the 27th term of the sequence.

Q9:

Consider the sequence −1,2,−3,4,−5,….

Write the sequence of the first 5 odd terms: 𝑎,𝑎,𝑎,𝑎,𝑎.

  • A1, 3, 5, 7, 9
  • B2, 4, 6, 8, 10
  • C − 1 , − 3 , − 5 , − 7 , − 9
  • D − 2 , − 4 , − 6 , − 8 , − 1 0
  • E − 1 , 2 , − 3 , 4 , − 5

Write a piecewise function that describes 𝑎.

  • A 𝑎 = 𝑛 − 1 𝑛 𝑛 + 1 𝑛 i f i s o d d i f i s e v e n
  • B 𝑎 = − 𝑛 𝑛 𝑛 𝑛 i f i s o d d i f i s e v e n
  • C 𝑎 = 𝑛 𝑛 𝑛 − 1 𝑛 i f i s o d d i f i s e v e n
  • D 𝑎 = 𝑛 + 1 𝑛 𝑛 𝑛 i f i s o d d i f i s e v e n
  • E 𝑎 = − 𝑛 𝑛 𝑛 + 1 𝑛 i f i s o d d i f i s e v e n

Q10:

Is the sequence (14,17,20,23,…) finite or infinite?

  • Afinite
  • Binfinite

Q11:

Is the sequence (−88,−87,−86,−85) finite or infinite?

  • Ainfinite
  • Bfinite

Q12:

Is the sequence (72,77,82,87,…,122) finite or infinite?

  • Ainfinite
  • Bfinite

Q13:

Is the sequence with general term 8𝑛−22, where 𝑛∈ℤ, finite or infinite?

  • Afinite
  • Binfinite

Q14:

Is the sequence with general term 5𝑛+73, where 𝑛∈ℤ, finite or infinite?

  • Afinite
  • Binfinite

Q15:

Find the range of the infinite arithmetic sequence represented in the figure below.

  • A { 0 , 1 , 2 , 3 , … }
  • B { 1 , 2 , 3 , 4 , … }
  • C { 0 , 1 , 2 , 3 }
  • D ℝ
  • E [ 0 , 9 ]

Q16:

Find the range of the infinite arithmetic sequence represented in the figure below.

  • A { − 3 , − 5 , − 7 , − 9 , … }
  • B { 1 , 2 , 3 , 4 , … }
  • C { − 3 , − 5 , − 7 , − 9 }
  • D ℝ
  • E [ − 9 , − 3 ]

Q17:

Is the sequence 𝑎=4𝑛+58 for 𝑛∈{1,2,3,4,5} finite or infinite?

  • Ainfinite
  • Bfinite

Q18:

What kind of sequence is the following: 1,110,1100,11,000,…?

  • ANeither geometric nor arithmetic
  • BGeometric only
  • CArithmetic only
  • DGeometric and arithmetic

Q19:

What kind of sequence is the following: 12,136,236,112,436,…?

  • Aarithmetic only
  • Bneither geometric nor arithmetic
  • Cgeometric and arithmetic
  • Dgeometric only

Q20:

Sophia has 223 pennies in her piggy bank. Each week after that, she puts in another 50 pennies.

What type of sequence do the successive amounts in her piggy bank make?

  • Aarithmetic only
  • Bneither geometric nor arithmetic
  • Cgeometric only
  • Dgeometric and arithmetic

List the first four terms of the sequence of the amounts starting with 223 as the first term.

  • A223, 173, 123, 73
  • B233, 283, 313, 343
  • C223, 273, 323, 373
  • D233, 203, 173, 143
  • E233, 293, 353, 413

For 𝑛≥1, let 𝐴(𝑛) be the number of pennies Sophia has after n weeks so that 𝐴(0)=223. What is 𝐴(4)?

Is 𝐴(6)>500 true? If so, express it in words without referring to the function A.

  • ANo, before six weeks, Sophia has less than 500 pennies in her piggy bank.
  • BYes, before six weeks, Sophia has more than 500 pennies in her piggy bank.
  • CYes, after six weeks, Sophia had more than 500 pennies in her piggy bank.
  • DNo, after six weeks, Sophia had less than 500 pennies in her piggy bank.
  • EYes, after six weeks, Sophia has 500 pennies in her piggy bank.

Q21:

Consider the sequence −1,2,−3,4,−5,6,….

What kind of sequence is this?

  • AGeometric and arithmetic
  • BArithmetic only
  • CNeither geometric nor arithmetic
  • DGeometric only

What are the 234th and the 1,371st members of this sequence?

  • A − 2 3 4 , − 1 , 3 7 1
  • B − 2 3 4 , 1 , 3 7 1
  • C 2 3 4 , − 1 , 3 7 1
  • D 234, 1,371

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 )
  • B 𝑛 ( − 1 )
  • C 𝑛 ( − 1 )
  • D ( − 1 )
  • E 𝑛 ( 1 )

Q22:

What kind of sequence is the following: 1,0.1,0.01,0.001,…?

  • Aneither geometric nor arithmetic
  • Bgeometric and arithmetic
  • Carithmetic only
  • Dgeometric only

Q23:

What kind of sequence is the following: 23,23,23,23,…?

  • Aneither geometric nor arithmetic
  • Bgeometric and arithmetic
  • Cgeometric only
  • Darithmetic only

Q24:

What kind of sequence is the following: 0.1,0.11,0.111,0.1111,…?

  • Ageometric and arithmetic
  • Bgeometric only
  • Carithmetic only
  • Dneither geometric nor arithmetic

Q25:

Consider the sequence 12,34,56,78,….

List the numbers at positions 5, 6, and 7.

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

What type of sequence is the given sequence?

  • Ageometric
  • Bneither geometric nor arithmetic
  • Carithmetic
  • Dgeometric and arithmetic

What is the number at position 13?

  • A 2 1 2 2
  • B 2 5 2 6
  • C 2 7 2 8
  • D 2 9 3 0
  • E 2 3 2 4

What is the number at position 𝑛?

  • A 𝑛 − 1 𝑛
  • B 2 𝑛 + 1 2 𝑛
  • C 2 𝑛 2 𝑛 − 1
  • D 2 𝑛 − 1 2 𝑛
  • E 𝑛 + 1 𝑛

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