Challenging +1.2 This is a structured induction proof with a given formula to verify. While it involves the golden ratio and Fibonacci numbers (conceptually interesting), the proof follows a standard template: verify base cases, assume for n and n-1, then show for n+1 using the recurrence relation and the property φ² = φ + 1. The algebraic manipulation is straightforward once the setup is recognized. Slightly above average due to the non-trivial formula and need to use φ's defining property, but well within reach for Further Maths students familiar with induction.
1 = 1 . + 0 = ( F ) + ( F ) so result true for n = 1
1 0
Assuming that k = ( F ) + ( F ) …
k k − 1
… it follows that 𝜙𝑘 + 1 = 𝜙((𝐹 )𝜙+(𝐹 ))
𝑘 𝑘 − 1
( ) ( ) ( )
F + 1 + F
=
k k − 1
=
( ) ( ) ( )
F +F + F = F +F
k k − 1 k k + 1 k
and result is also true for n = k + 1
Since true for n = 1 and (true for n = k true for n = k +
Answer
Marks
1), the result follows for all n 1 by induction
B1
M1
M1
M1
A1*
A1dep
Answer
Marks
[6]
2.5
1.2
1.1a
3.1a
2.2a
Answer
Marks
2.4
Induction hypothesis explicitly stated (so “assume the
result is true for n = k” does not suffice without a clear
indication as to what it is that is being assumed)
k + 1
Attempt at with explicit use of induction
hypothesis
Use of result for n = 2 or from known property of
(e.g. from the auxiliary equation of the Fib. sequence)
or by direct calculation
(k + 1)th case established clearly from use of the
defining Fibonacci sequence property
Explanation of the inductive logic (can only be
awarded if the result has actually been proven).
Question 3:
3 | 1 = 1 . + 0 = ( F ) + ( F ) so result true for n = 1
1 0
Assuming that k = ( F ) + ( F ) …
k k − 1
… it follows that 𝜙𝑘 + 1 = 𝜙((𝐹 )𝜙+(𝐹 ))
𝑘 𝑘 − 1
( ) ( ) ( )
F + 1 + F
=
k k − 1
=
( ) ( ) ( )
F +F + F = F +F
k k − 1 k k + 1 k
and result is also true for n = k + 1
Since true for n = 1 and (true for n = k true for n = k +
1), the result follows for all n 1 by induction | B1
M1
M1
M1
A1*
A1dep
[6] | 2.5
1.2
1.1a
3.1a
2.2a
2.4 | Induction hypothesis explicitly stated (so “assume the
result is true for n = k” does not suffice without a clear
indication as to what it is that is being assumed)
k + 1
Attempt at with explicit use of induction
hypothesis
Use of result for n = 2 or from known property of
(e.g. from the auxiliary equation of the Fib. sequence)
or by direct calculation
(k + 1)th case established clearly from use of the
defining Fibonacci sequence property
Explanation of the inductive logic (can only be
awarded if the result has actually been proven).