Mathematical-Reasoning · Unit 14 · Video 2 · Interactive Practice

Fibonacci Numbers: Patterns, Rabbits and Sanskrit Verse

IKey Formulas

Statement	Name	Status
$f_1 = 1$ , $f_2 = 1$ , and $f_{n+2} = f_{n+1} + f_n$ for each $n \in \mathbb{N}$	Fibonacci numbers	Definition
$f_1 + f_2 + \cdots + f_{n-1} = f_{n+1} - 1$ for each $n \ge 2$	Sum pattern	Conjecture
$f_1^2 + f_2^2 + \cdots + f_n^2 = f_n f_{n+1}$ for each $n \in \mathbb{N}$	Sum of squares	Conjecture
$f_{n-1} f_{n+1} - f_n^2 = (-1)^n$ for each $n \ge 2$	Cassini's identity	Conjecture

Key Insight: A table checks finitely many terms, so every pattern found in $f_1$ to $f_{20}$ is only a conjecture about all $n \in \mathbb{N}$ ; the divisibility, sum, squares and neighbour patterns can be proved by induction.

IIDivisibility Lives in the Index

Which of $f_1$ to $f_{20}$ does a given number divide, and what do their indices have in common?

Divisor d

IIIRatios of Consecutive Terms

The ratios $f_{n+1}/f_n$ fall and rise in turn, and the swings shrink near $1.618$ .

n 4

💡 Settling near $1.618$ is a claim about a limit: no finite list of ratios can establish it, and proving it takes more than induction alone.

IVHemachandra's Syllable Patterns

Every pattern of length $n$ ends in a short or a long syllable, so $h_n = h_{n-1} + h_{n-2}$ for $n \ge 3$ .

Length n 5

VQuiz Questions

Problem 1 · Run the Recurrence

Given: $f_{10} = 55$ and $f_{11} = 89$ — find $f_{13}$ .

✅ Correct! Two additions:

f_{12} = 89 + 55 = 144

, then

f_{13} = 144 + 89 = 233

❌ One step short.

89 + 55 = 144

f_{12}

; one more addition is needed to reach

f_{13}

❌ Not a Fibonacci step.

178 = 2 \cdot 89

doubles the last term; the rule adds the two terms before,

f_{11} + f_{10}

❌ Wrong pair of summands.

199 = 144 + 55 = f_{12} + f_{10}

; the rule uses the two terms immediately before:

f_{13} = f_{12} + f_{11}

Show solution

Apply $f_{n+2} = f_{n+1} + f_n$ twice:

f_{12} = f_{11} + f_{10} = 89 + 55 = 144

f_{13} = f_{12} + f_{11} = 144 + 89 = 233

Problem 2 · The Sum Pattern's Indices

Given: the conjecture $f_1 + f_2 + \cdots + f_{n-1} = f_{n+1} - 1$ for $n \ge 2$ , and $f_{11} = 89$ , $f_{12} = 144$ , $f_{13} = 233$ — find $f_1 + f_2 + \cdots + f_{10}$ as the conjecture predicts.

✅ Correct! The last summand is

f_{n-1} = f_{10}

, so

n = 11

and the sum is

f_{12} - 1 = 143

; adding the ten terms directly gives

143

too.

❌ Index off by one.

88 = f_{11} - 1

treats the last summand as

f_n

; in the conjecture it is

f_{n-1}

, so

f_{10}

means

n = 11

and the right side is

f_{12} - 1

❌ Missing the $-1$ .

f_{12} = 144

is one more than the sum; the conjecture says the sum is exactly one less than

f_{n+1}

❌ Index off by one the other way.

232 = f_{13} - 1

would be the sum through

f_{11}

; here the sum stops at

f_{10}

, so

n = 11

and

n + 1 = 12

Show solution

The last summand is $f_{n-1}$ ; setting $n - 1 = 10$ gives $n = 11$ . The conjecture predicts

f_1 + f_2 + \cdots + f_{10} = f_{12} - 1 = 144 - 1 = 143.

Check by direct addition: $1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 = 143$ . This confirms the case $n = 11$ only; it is still not a proof for every $n$ .

Problem 3 · Neighbours Without Multiplying

Given: $f_{10} = 55$ and Cassini's conjecture $f_{n-1} f_{n+1} - f_n^2 = (-1)^n$ — find the difference $f_9 f_{11} - f_{10}^2$ , then the product $f_9 f_{11}$ .

What is $f_9 f_{11} - f_{10}^2$ ?

What is $f_9 f_{11}$ ?

✅ Correct! With

n = 10

even, the product is one more than the square:

3025 + 1 = 3026

, and indeed

34 \cdot 89 = 3026

❌ Wrong sign.

-1

is the value when the middle index is odd; here the middle term is

f_{10}

, so

n = 10

and

(-1)^{10} = 1

❌ They are never equal. In every case checked, the neighbour product differs from the square by exactly

1

; the conjecture gives

(-1)^n

, never

0

❌ Sign flipped.

3024 = 3025 - 1

would follow from a difference of

-1

; for even

n = 10

the difference is

+1

❌ That is the square itself.

3025 = 55^2 = f_{10}^2

; the neighbour product differs from it by

1

❌ Not quite. Substitute

n = 10

into

(-1)^n

❌ Not quite. Solve

f_9 f_{11} - 3025 = 1

for the product.

Show solution

The middle term is $f_{10}$ , so take $n = 10$ in the conjecture:

f_9 f_{11} - f_{10}^2 = (-1)^{10} = 1.

With $f_{10}^2 = 55^2 = 3025$ :

f_9 f_{11} = 3025 + 1 = 3026.

Check: $f_9 = 34$ and $f_{11} = 89$ , and $34 \cdot 89 = 3026$ .

Problem 4 · Syllable Patterns of Length 8

Given: a short syllable has length $1$ , a long one length $2$ , and $h_n$ counts the patterns of total length $n$ , with $h_1 = 1$ , $h_2 = 2$ and $h_{n+2} = h_{n+1} + h_n$ — find $h_8$ .

✅ Correct!

h_n = f_{n+1}

for every term listed, and

h_8 = f_9 = 34

❌ Missed the shift.

21 = f_8

; the syllable counts start

1, 2

rather than

1, 1

, so they run one place ahead:

h_8 = f_9

❌ Shifted two places.

55 = f_{10} = h_9

, the count for length

9

; length

8

is one step earlier.

❌ Too many.

2^8

counts strings of

8

syllables, each short or long; but a long syllable uses

2

units, so most such strings are longer than

8

Show solution

Run the recurrence from $h_1 = 1$ , $h_2 = 2$ :

h_3 = 3,\quad h_4 = 5,\quad h_5 = 8,\quad h_6 = 13,\quad h_7 = 21,\quad h_8 = h_7 + h_6 = 21 + 13 = 34.

Of the $34$ patterns, $h_7 = 21$ end in a short syllable and $h_6 = 13$ end in a long one. The list matches $f_2, f_3, \ldots$ , one place ahead of the Fibonacci numbers: $h_8 = f_9 = 34$ .

Solved: 0 / 4