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Fibonacci sequence inquiry

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Inquiry prompt

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Link to Regulatory cards

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Exploration

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True or false?

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Work out the first 30 terms of the Fibonacci sequence.

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Work out the first 30 terms of the Fibonacci sequence.

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Summing terms of the Fibonacci sequence

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Acknowledgement Don Steward

… the sum any three consecutive terms?
… the sum of the first four of any five consecutive terms and the sum of the third and fifth terms?
… half the sum of the first and fourth terms of any four consecutive terms?
… a third of the sum of the first and fifth terms of any five consecutive terms?
… a third of the sum of the third and seventh terms of any seven consecutive terms?
… a seventh of the sum of the first and ninth terms of any nine consecutive terms?

Sums of consecutive terms in the Fibonacci sequence

What do you notice about …

… a quarter of the sum of any six consecutive terms?
… an eleventh of the sum of any ten consecutive terms?
… the sums of the first two, first three, first four, etc, terms of the Fibonacci sequence?
… half the difference of the ninth and sixth terms of any nine consecutive terms?
… half the difference of the eighth and fifth terms of any eight consecutive terms?
… a seventh of the sum of the third and eleventh terms of any eleven consecutive terms?

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True or false?

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True or false?

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Algebraic proof

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Work out the first 20 algebraic terms of the Fibonacci sequence.

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Work out the first 20 algebraic terms of the Fibonacci sequence.

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True or false?

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Proof

Prove algebraically some of the other results you have found about the sums of terms in the Fibonacci sequence.

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Proof by induction

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To prove

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To prove

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To prove

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Another proof

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Fibonacci-type sequences

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Fill in the missing terms in each sequence.

A

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Fill in the missing terms in each sequence.

B

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Fill in the missing terms in each sequence.

C

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Fill in the missing terms in each sequence.

D

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Solutions

A

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B

Solutions

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C

Solutions

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D

Solutions

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Fibonacci-type sequences

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Fibonacci-type sequences

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Forming and solving

simultaneous equations

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Place two numbers in different cells.

Is it always possible to find the rest of the sequence?
What if the second number was less than the first?

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