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Warm UP!. Identify the following as Arithmetic, Geometric, or neither: 2, 7, 12, 17, … 2. Find the nth term for the sequence: 2, 20, 200, 2000, … 3. Generate the first four terms of the sequence given its recursive formula: a 1 = 6 and a n = 2a n-1 + 3

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Warm UP!

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Warm up

Warm UP!

Identify the following as Arithmetic, Geometric, or neither: 2, 7, 12, 17, …

2. Find the nth term for the sequence: 2, 20, 200, 2000, …

3. Generate the first four terms of the sequence given its recursive formula: a1 = 6 and an = 2an-1 + 3

4. Find the 150th term of the sequence an = 0.5n + 8

Quiz Time


Warm up

Graphical interpretation of limits for explicit sequences investigationComplete the task. You may work with a partner. You have 20 minutes.

LG 6-2: Limits of Sequences


Warm up

  • A limit is like asking the following question: “What happens to a sequence an when n approaches infinity?”

  • A sequence can behave in two ways:

    • Convergence

    • Divergence


Examples of converging sequences

Examples of converging sequences

The sequence converges to a unique value: 0

We write this as


Examples of converging sequences1

Examples of converging sequences

The sequence converges to a unique value: 1

We write this as


Examples of diverging sequences

Examples of diverging sequences

This sequence diverges. You can see it is going up – to infinity.

We can write this as


Convergence or divergence

Convergence or Divergence?

Given a sequence an there are several possibilities as to its convergence behavior:

  • The sequence may converge to a number or to 0.

  • The sequence may diverge in one of three ways:

    • To positive infinity – “goes to infinity”

    • To negative infinity – “goes to negative infinity”

    • The limit might not exist at all (DNE)


Limits for rational sequences

Limits for rational sequences

  • Converge

    • If the degrees are the same, the limit is equal to the ratio of the leading coefficients

    • If the degree of the denominator is larger than the numerator, the limit is 0.

  • Diverge

    • If the degree of the numerator is larger than the denominator, the limit goes to infinity


Look at numbers 12 16

Look at numbers 12 – 16

When a geometric sequence is in explicit form, you only need to use the common ratio to determine the limit as the sequence approaches infinity.

The geometric sequence a1(r)n-1 is divergent if |r| > 1 or r > 1 and converges to 0 if |r| < 1.


Limits practice

Limits Practice:

5. 8, -5, 8, -5, 8,. . .

Estimate the limits:

7. 8.


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