Fibonacci and Related Sequences - Sequences and Series

OCR Further Additional Pure AS 2018 June Q6

8 marks Challenging +1.8

6 The Fibonacci sequence $\left\{ F _ { n } \right\}$ is defined by $F _ { 0 } = 0 , F _ { 1 } = 1$ and $F _ { n } = F _ { n - 1 } + F _ { n - 2 }$ for all $n \geqslant 2$.

Show that $F _ { n + 5 } = 5 F _ { n + 1 } + 3 F _ { n }$
Prove that $F _ { n }$ is a multiple of 5 when $n$ is a multiple of 5 .

OCR Further Additional Pure AS 2022 June Q3

7 marks Challenging +1.2

3 The sequence $\left\{ U _ { n } \right\}$ is given by $U _ { 1 } = 0 , U _ { 2 } = - 1$ and $U _ { n + 2 } = U _ { n + 1 } + U _ { n } + n - 1$ for $n \geqslant 1$.

List the first seven terms of this sequence. The Fibonacci sequence $\left\{ \mathrm { F } _ { \mathrm { n } } \right\}$ is given by $\mathrm { F } _ { 1 } = 1 , \mathrm {~F} _ { 2 } = 1$ and $\mathrm { F } _ { \mathrm { n } + 2 } = \mathrm { F } _ { \mathrm { n } + 1 } + \mathrm { F } _ { \mathrm { n } }$ for $n \geqslant 1$.
1. By comparing the two sequences, give the relationship between $\mathrm { U } _ { \mathrm { n } }$ and $\mathrm { F } _ { \mathrm { n } }$.
2. Show that the relationship found in part (b)(i) holds for all $n \geqslant 1$.

OCR Further Additional Pure 2023 June Q7

10 marks Challenging +1.8

7 Binet's formula for the $n$th Fibonacci number is given by $\mathrm { F } _ { \mathrm { n } } = \frac { 1 } { \sqrt { 5 } } \left( \alpha ^ { \mathrm { n } } - \beta ^ { \mathrm { n } } \right)$ for $n \geqslant 0$, where $\alpha$ and $\beta$ (with $\alpha > 0 > \beta$ ) are the roots of $x ^ { 2 } - x - 1 = 0$.

Write down the values of $\alpha + \beta$ and $\alpha \beta$.
Consider the sequence $\left\{ \mathrm { S } _ { \mathrm { n } } \right\}$, where $\mathrm { S } _ { \mathrm { n } } = \alpha ^ { \mathrm { n } } + \beta ^ { \mathrm { n } }$ for $n \geqslant 0$.
1. Determine the values of $S _ { 2 }$ and $S _ { 3 }$.
2. Show that $S _ { n + 2 } = S _ { n + 1 } + S _ { n }$ for $n \geqslant 0$.
3. Deduce that $S _ { n }$ is an integer for all $n \geqslant 0$.
A student models the terms of the sequence $\left\{ \mathrm { S } _ { \mathrm { n } } \right\}$ using the formula $\mathrm { T } _ { \mathrm { n } } = \alpha ^ { \mathrm { n } }$.
1. Explain why this formula is unsuitable for every $n \geqslant 1$.
2. Considering the cases $n$ even and $n$ odd separately, state a modification of the formula $\mathrm { T } _ { \mathrm { n } } = \alpha ^ { \mathrm { n } }$, other than $\mathrm { T } _ { \mathrm { n } } = \alpha ^ { \mathrm { n } } + \beta ^ { \mathrm { n } }$, such that $\mathrm { T } _ { \mathrm { n } } = \mathrm { S } _ { \mathrm { n } }$ for all $n \geqslant 1$.

Pre-U Pre-U 9795/1 2016 June Q11

5 marks Challenging +1.8

11

The sequence of Fibonacci Numbers $\left\{ F _ { n } \right\}$ is given by $$F _ { 1 } = 1 , \quad F _ { 2 } = 1 \quad \text { and } \quad F _ { n + 1 } = F _ { n } + F _ { n - 1 } \text { for } n \geqslant 2 .$$ Write down the values of $F _ { 3 }$ to $F _ { 6 }$.
The sequence of functions $\left\{ \mathrm { p } _ { n } ( x ) \right\}$ is given by $$\mathrm { p } _ { 1 } ( x ) = x + 1 \quad \text { and } \quad \mathrm { p } _ { n + 1 } ( x ) = 1 + \frac { 1 } { \mathrm { p } _ { n } ( x ) } \text { for } n \geqslant 1$$
1. Find $\mathrm { p } _ { 2 } ( x )$ and $\mathrm { p } _ { 3 } ( x )$, giving each answer as a single algebraic fraction, and show that $\mathrm { p } _ { 4 } ( x ) = \frac { 3 x + 5 } { 2 x + 3 }$.
2. Conjecture an expression for $\mathrm { p } _ { n } ( x )$ as a single algebraic fraction involving Fibonacci numbers, and prove it by induction for all integers $n \geqslant 2$.