Understanding Sequences: Boundedness, Monotonicity, and Convergence

Understanding Sequences: Boundedness, Monotonicity, and Convergence

Assessment

Interactive Video

Mathematics

9th - 12th Grade

Practice Problem

Hard

CCSS
HSF.BF.A.2, HSF-IF.C.8B

Standards-aligned

Created by

Amelia Wright

FREE Resource

Standards-aligned

CCSS.HSF.BF.A.2
,
CCSS.HSF-IF.C.8B
The video tutorial explores the sequence a_n = (4^n)/(5^n), examining its properties. It discusses whether the sequence is bounded or unbounded, monotonic or not, and whether it converges or diverges. The sequence is identified as a geometric sequence with a common ratio less than one, making it bounded and monotonic. The tutorial concludes that the sequence converges to zero, supported by a theorem on bounded and monotonic sequences.

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10 questions

Show all answers

1.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

What is the initial form of the sequence a_n discussed in the video?

a_n = (4n)/(5n)

a_n = (5^n)/(4^n)

a_n = (n^4)/(n^5)

a_n = (4^n)/(5^n)

Tags

CCSS.HSF.BF.A.2

2.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

What does it mean for a sequence to be bounded?

It has no bounds.

It has both an upper and a lower bound.

It has only a lower bound.

It has only an upper bound.

Tags

CCSS.HSF.BF.A.2

3.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

What are the bounds of the sequence a_n = (4^n)/(5^n)?

Bounded above by 1 and below by 1/5

Bounded above by 1 and below by 0

Bounded above by 4/5 and below by 0

Bounded above by 5/4 and below by 1

Tags

CCSS.HSF.BF.A.2

4.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

What characterizes a monotonically decreasing sequence?

Each term is less than or equal to the previous term.

Each term is greater than the previous term.

Each term is greater than or equal to the previous term.

Each term is equal to the previous term.

Tags

CCSS.HSF.BF.A.2

5.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

Why is the sequence a_n = (4^n)/(5^n) considered monotonic?

It alternates between increasing and decreasing.

It is decreasing with each term.

It remains constant.

It is increasing with each term.

Tags

CCSS.HSF-IF.C.8B

6.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

What theorem is used to determine the convergence of the sequence?

A sequence that is bounded and monotonic will diverge.

A sequence that is bounded and monotonic will converge.

A sequence that is unbounded and non-monotonic will diverge.

A sequence that is unbounded and monotonic will converge.

7.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

To what value does the sequence a_n = (4^n)/(5^n) converge?

Infinity

0

4/5

1

Tags

CCSS.HSF-IF.C.8B

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