Group 1 Names: Xavier, Sam T., Katie S., Tamarr S., Ari Z.

Consider the set of all increasing sequences:

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1. Some increasing sequences converge to a real number. What property do these sequences all have in common?

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• What happens to an increasing sequence that does not converge to a real number?

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• Complete the following if statements to generate a conjecture that classifies all increasing sequences:

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Let be an increasing sequence.

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If is ______________________ then converges to a real number.

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If is _______________________then tends to infinity.

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• Are there any other possibilities for an increasing sequence? Is it possible for an increasing sequence to not converge to a real number AND also not tend to infinity?

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Bounded above

unbounded

The function is continuous and converge to the root.

The sequence is discontinuous (diverges) when it does not converge to a real number. It might also tend to infinity.

Imaginary roots?

Group 2 Names: Konrad B, Paige B, Noah S, Poem O, Nicholas H

Consider the set of all increasing sequences:

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• Some increasing sequences converge to a real number. What property do these sequences all have in common?

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​

​

• What happens to an increasing sequence that does not converge to a real number?

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• Complete the following if statements to generate a conjecture that classifies all increasing sequences:

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Let be an increasing sequence.

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If is ______________________ then converges to a real number.

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If is _______________________then tends to infinity.

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• Are there any other possibilities for an increasing sequence? Is it possible for an increasing sequence to not converge to a real number AND also not tend to infinity?

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Bounded above

Unbounded

The sequence has an upper bound.

The sequence tends to infinity.

No, if an increasing sequence does not converge to a real number, it must tend to infinity (and vice versa) by our choices for problem 3.

Group 3 Names: Amna , Beau, Kayla, Safia, Patricia

Consider the set of all increasing sequences:

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• Some increasing sequences converge to a real number. What property do these sequences all have in common?

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​

​

• What happens to an increasing sequence that does not converge to a real number?

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​

• Complete the following if statements to generate a conjecture that classifies all increasing sequences:

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Let be an increasing sequence.

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If is ______________________ then converges to a real number.

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If is _______________________then tends to infinity.

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• Are there any other possibilities for an increasing sequence? Is it possible for an increasing sequence to not converge to a real number AND also not tend to infinity?

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continuous

Divergent?/discontin

In common property would be convergent to a root. Might have a upper bound

The sequence would go to +infinity or be divergent.

NO

Group 4 Names: Sam Roberts, Alejandra Acevedo, Jack Lutz

Consider the set of all increasing sequences:

​

• Some increasing sequences converge to a real number. What property do these sequences all have in common?

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​

​

​

​

• What happens to an increasing sequence that does not converge to a real number?

​

​

​

​

​

• Complete the following if statements to generate a conjecture that classifies all increasing sequences:

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Let be an increasing sequence.

​

If is ______________________ then converges to a real number.

​

If is _______________________then tends to infinity.

​

• Are there any other possibilities for an increasing sequence? Is it possible for an increasing sequence to not converge to a real number AND also not tend to infinity?

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bounded

unbounded

The sequence must be bounded above. The sequence must not increase infinitely. The sequence either increases by smaller and smaller amounts, or it is eventually constant.

Values of the function head off in different directions.

Or It is unbounded.

Or It must go to infinity.

No, for all increasing sequences, they either converge to a real number or tend to infinity.

Group 5 Names: Tim Swanson, Kelby Reynolds, Eric Phillips, Davis Pies, Aaron Richard

Consider the set of all increasing sequences:

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• Some increasing sequences converge to a real number. What property do these sequences all have in common?

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​

​

​

• What happens to an increasing sequence that does not converge to a real number?

​

​

​

​

​

• Complete the following if statements to generate a conjecture that classifies all increasing sequences:

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Let be an increasing sequence.

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If is ______________________ then converges to a real number.

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If is _______________________then tends to infinity.

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• Are there any other possibilities for an increasing sequence? Is it possible for an increasing sequence to not converge to a real number AND also not tend to infinity?

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Bounded above

Not convergent

Bounded above

Tends towards infinity

HMMMMMMMMMMMM

Group X Names: Benjamin Lutz, Eric Zmitrovich, Matt Little, Koby Scheetz

Consider the set of all increasing sequences:

​

• Some increasing sequences converge to a real number. What property do these sequences all have in common?

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​

​

​

​

• What happens to an increasing sequence that does not converge to a real number?

​

​

​

​

​

• Complete the following if statements to generate a conjecture that classifies all increasing sequences:

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Let be an increasing sequence.

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If is ______________________ then converges to a real number.

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If is _______________________then tends to infinity.

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• Are there any other possibilities for an increasing sequence? Is it possible for an increasing sequence to not converge to a real number AND also not tend to infinity?

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Eventually Constant

a_n < a_n+1 for n ∊ N

Eventually Constant (maybe not required)

Tends to Infinity and Unbounded Above

This is not possible.