Worksheet: Monotonic Sequences

In this worksheet, we will practice determining monotonic sequences.

Q1:

Is the sequence 𝑇 = 1 4 𝑛 𝑛 + 6 increasing, decreasing, or neither?

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

Q2:

The 𝑛 t h term of a sequence is 𝑇 𝑛 . If 𝑇 = 𝑛 𝑛 2 , is 𝑇 > 𝑇 𝑛 + 1 𝑛 ?

  • Ayes
  • Bno

Q3:

Use < , = , or > to complete the following: A sequence with 𝑛 t h term 𝑇 𝑛 is constant if 𝑇 𝑇 𝑛 + 1 𝑛 for each 𝑛 1 .

  • A <
  • B >
  • C =

Q4:

Use < , = , or > to complete the following: A sequence with 𝑛 t h term 𝑇 𝑛 is strictly increasing if 𝑇 𝑇 𝑛 + 1 𝑛 for each 𝑛 1 .

  • A <
  • B =
  • C >

Q5:

Is the sequence 𝑇 = 2 3 𝑛 + 1 9 𝑛 increasing, decreasing, or neither?

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

Q6:

Is the sequence 𝑇 = 2 8 2 9 2 8 𝑛 𝑛 2 3 , where 𝑛 1 increasing, decreasing, or neither?

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

Q7:

Is the sequence 𝑇 = 1 1 𝑛 + 4 4 𝑛 2 increasing, decreasing, or neither?

  • Adecreasing
  • Bneither increasing nor decreasing
  • Cincreasing

Q8:

Is the sequence 𝑇 = 9 𝑛 3 2 𝑛 increasing, decreasing, or neither?

  • Aneither increasing nor decreasing
  • Bdecreasing
  • Cincreasing

Q9:

Is the sequence 𝑇 = ( 3 1 ) 𝑛 𝑛 increasing, decreasing or neither?

  • Adecreasing
  • Bincreasing
  • Cneither increasing nor decreasing

Q10:

Is the sequence 𝑇 = 1 1 9 𝑛 1 6 𝑛 increasing, decreasing, or neither?

  • Aneither increasing nor decreasing
  • Bincreasing
  • Cdecreasing

Q11:

The 𝑛 t h term of a sequence is 𝑇 𝑛 . If 𝑇 = 𝑛 𝑛 2 , is 𝑇 > 𝑇 𝑛 𝑛 + 1 ?

  • Ano
  • Byes

Q12:

Is the sequence 𝑇 = 1 1 + 2 𝑛 𝑛 increasing, decreasing, or neither?

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

Q13:

Use < , = , or > to complete the following: A sequence with 𝑛 t h term 𝑇 𝑛 is strictly decreasing if 𝑇 𝑇 𝑛 + 1 𝑛 for each 𝑛 1 .

  • A =
  • B >
  • C <

Q14:

A geometric sequence has first term 𝑎 and common ratio 𝑟 . In which of the following cases will the sequence be decreasing?

  • A 𝑎 < 0 , 1 < 𝑟 < 0
  • B 𝑎 > 0 , 1 < 𝑟 < 0
  • C 𝑎 < 0 , 0 < 𝑟 < 1
  • D 𝑎 > 0 , 0 < 𝑟 < 1
  • E 𝑎 > 1 , 1 < 𝑟 < 0

Q15:

Consider the sequence for .

Is ?

  • Ano
  • Byes

Is ?

  • Ayes
  • Bno

Define and by and . Write in simplified form.

  • A
  • B
  • C
  • D
  • E

Using the above and the quadratic formula, find the smallest integer so that whenever .

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