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Worksheet: Monotonic Sequences

Q1:

Consider the sequence π‘Ž = 3 𝑛 βˆ’ 4 2 𝑛 βˆ’ 1 𝑛 2 for 𝑛 β‰₯ 1 .

Is π‘Ž > π‘Ž 1 2 ?

  • Ano
  • Byes

Is π‘Ž > π‘Ž 3 4 ?

  • Ayes
  • Bno

Define 𝑃 , 𝑄 , 𝑅 , and 𝑆 by π‘Ž = 𝑃 𝑄 𝑛 and π‘Ž = 𝑅 𝑆 𝑛 + 1 . Write 𝑃 𝑆 βˆ’ 𝑄 𝑅 in simplified form.

  • A 𝑛 βˆ’ 3 𝑛 βˆ’ 5 2
  • B 6 𝑛 βˆ’ 1 0 𝑛 βˆ’ 7 2
  • C 6 𝑛 βˆ’ 1 0 𝑛 βˆ’ 5 2
  • D 𝑛 + 3 𝑛 βˆ’ 5 2
  • E 6 𝑛 + 1 0 𝑛 βˆ’ 7 2

Using the above and the quadratic formula, find the smallest integer 𝑁 so that π‘Ž > π‘Ž 𝑛 𝑛 + 1 whenever 𝑛 β‰₯ 𝑁 .

Q2:

Is the sequence increasing, decreasing, or neither?

  • A is increasing.
  • B is decreasing.
  • C is neither increasing nor decreasing.

Q3:

The term of a sequence is . If , is ?

  • Ayes
  • Bno

Q4:

Use , , or to complete the following: A sequence with term is constant if for each .

  • A
  • B
  • C

Q5:

Use < , = , or > to complete the following: A sequence with 𝑛 t h term π‘Ž 𝑛 is strictly increasing if π‘Ž π‘Ž 𝑛 + 1 𝑛 for each 𝑛 β‰₯ 1 .

  • A <
  • B =
  • C >

Q6:

Is the sequence increasing, decreasing, or neither?

  • A is neither increasing nor decreasing.
  • B is increasing.
  • C is decreasing.

Q7:

Is the sequence , where increasing, decreasing, or neither?

  • A is neither increasing nor decreasing.
  • B is decreasing.
  • C is increasing.

Q8:

Is the sequence increasing, decreasing, or neither?

  • Adecreasing
  • Bneither increasing nor decreasing
  • Cincreasing

Q9:

Is the sequence increasing, decreasing, or neither?

  • Aneither increasing nor decreasing
  • Bdecreasing
  • Cincreasing

Q10:

Is the sequence increasing, decreasing or neither?

  • Adecreasing
  • Bincreasing
  • Cneither increasing nor decreasing

Q11:

Is the sequence increasing, decreasing, or neither?

  • Aneither increasing nor decreasing
  • Bincreasing
  • Cdecreasing

Q12:

The term of a sequence is . If , is ?

  • Ano
  • Byes

Q13:

Is the sequence increasing, decreasing, or neither?

  • A is neither increasing nor decreasing.
  • B is decreasing.
  • C is increasing.

Q14:

Use , , or to complete the following: A sequence with term is strictly decreasing if for each .

  • A
  • B
  • C

Q15:

A geometric sequence has first term π‘Ž and common ratio π‘Ÿ . In which of the following cases will the sequence be decreasing?

  • A π‘Ž < 0 , βˆ’ 1 < π‘Ÿ < 0
  • B π‘Ž > 0 , βˆ’ 1 < π‘Ÿ < 0
  • C π‘Ž < 0 , 0 < π‘Ÿ < 1
  • D π‘Ž > 0 , 0 < π‘Ÿ < 1
  • E π‘Ž > 1 , βˆ’ 1 < π‘Ÿ < 0