Question 2 - A-Level Maths

Edexcel FD2 2024 June — Question 2 3 marks

Exam Board	Edexcel
Module	FD2 (Further Decision 2)
Year	2024
Session	June
Marks	3
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Sequences and series, recurrence and convergence
Type	Finding constants from given sum formula
Difficulty	Standard +0.8 This question requires students to substitute the general solution into the recurrence relation and equate coefficients of powers of n and the constant term. While the algebraic manipulation is straightforward, it demands careful systematic work with multiple terms (exponential and polynomial) and solving a system to find four unknowns. It's above average difficulty due to the reverse-engineering nature and need for methodical coefficient comparison, but remains accessible to well-prepared Further Maths students.
Spec	4.04e Line intersections: parallel, skew, or intersecting

2. The general solution of the first order recurrence relation $$u _ { n + 1 } + a u _ { n } = b n ^ { 2 } + c n + d \quad n \geqslant 0$$ is given by $$u _ { n } = A ( 3 ) ^ { n } + 5 n ^ { 2 } + 1$$ where $A$ is an arbitrary non-zero constant.
By considering expressions for $u _ { n + 1 }$ and $u _ { n }$, find the values of the constants $a , b , c$ and $d$.

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Question 2:

Answer	Marks	Guidance
Answer	Mark	Guidance
$u_{n+1} = A(3)^{n+1} + 5(n+1)^2 + 1$	B1	Any correct expression for $u_{n+1}$
$u_{n+1} = A(3)^n(3) + 5(n+1)^2 + 1 \Rightarrow u_{n+1} = 3(u_n - 5n^2 - 1) + 5(n+1)^2 + 1$	M1	Eliminating $A$ to form a first order recurrence relation containing $u_{n+1}$ and $u_n$
$u_{n+1} - 3u_n = -10n^2 + 10n + 3$	A1	CAO; need not explicitly state $a=-3, b=-10, c=10, d=3$

Alternative 1:

Answer	Marks	Guidance
Answer	Mark	Guidance
$u_{n+1} + au_n = 0$, $u_{n+1} = -au_n$, C.F. $u_n = A(-a)^n$, $-a = 3 \Rightarrow a = -3$	B1	Considers the C.F. and deduces $a = -3$ with no other values stated
Try $\lambda n^2 + \mu n + \nu \Rightarrow \lambda = 5, \mu = 0, \nu = 1$	M1	Obtains P.S. and forms equation using $u_{n+1} \pm 3u_n$
$5(n+1)^2 + 1 - 3(5n^2+1) = bn^2 + cn + d \Rightarrow b = -10, c = 10, d = 3$	A1	CAO for $u_{n+1} - 3u_n = -10n^2 + 10n + 3$

Alternative 2:

Answer	Marks	Guidance
Answer	Mark	Guidance
$u_{n+1} + au_n = 3A(3)^n + 5n^2 + 10n + 6 + aA(3)^n + 5an^2 + a$	B1	Forms the correct equation for $u_{n+1} + au_n$
$3 + a = 0 \Rightarrow a = -3$; $5 + 5a = b \Rightarrow b = -10$; $c = 10$; $6 + a = d \Rightarrow d = 3$	M1	Attempts to compare coefficients – at least three terms seen
CAO	A1

# Question 2:

| Answer | Mark | Guidance |
|--------|------|----------|
| $u_{n+1} = A(3)^{n+1} + 5(n+1)^2 + 1$ | B1 | Any correct expression for $u_{n+1}$ |
| $u_{n+1} = A(3)^n(3) + 5(n+1)^2 + 1 \Rightarrow u_{n+1} = 3(u_n - 5n^2 - 1) + 5(n+1)^2 + 1$ | M1 | Eliminating $A$ to form a first order recurrence relation containing $u_{n+1}$ and $u_n$ |
| $u_{n+1} - 3u_n = -10n^2 + 10n + 3$ | A1 | CAO; need not explicitly state $a=-3, b=-10, c=10, d=3$ |

**Alternative 1:**

| Answer | Mark | Guidance |
|--------|------|----------|
| $u_{n+1} + au_n = 0$, $u_{n+1} = -au_n$, C.F. $u_n = A(-a)^n$, $-a = 3 \Rightarrow a = -3$ | B1 | Considers the C.F. and deduces $a = -3$ with no other values stated |
| Try $\lambda n^2 + \mu n + \nu \Rightarrow \lambda = 5, \mu = 0, \nu = 1$ | M1 | Obtains P.S. and forms equation using $u_{n+1} \pm 3u_n$ |
| $5(n+1)^2 + 1 - 3(5n^2+1) = bn^2 + cn + d \Rightarrow b = -10, c = 10, d = 3$ | A1 | CAO for $u_{n+1} - 3u_n = -10n^2 + 10n + 3$ |

**Alternative 2:**

| Answer | Mark | Guidance |
|--------|------|----------|
| $u_{n+1} + au_n = 3A(3)^n + 5n^2 + 10n + 6 + aA(3)^n + 5an^2 + a$ | B1 | Forms the correct equation for $u_{n+1} + au_n$ |
| $3 + a = 0 \Rightarrow a = -3$; $5 + 5a = b \Rightarrow b = -10$; $c = 10$; $6 + a = d \Rightarrow d = 3$ | M1 | Attempts to compare coefficients – at least three terms seen |
| CAO | A1 | |

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2. The general solution of the first order recurrence relation

$$u _ { n + 1 } + a u _ { n } = b n ^ { 2 } + c n + d \quad n \geqslant 0$$

is given by

$$u _ { n } = A ( 3 ) ^ { n } + 5 n ^ { 2 } + 1$$

where $A$ is an arbitrary non-zero constant.\\
By considering expressions for $u _ { n + 1 }$ and $u _ { n }$, find the values of the constants $a , b , c$ and $d$.\\

\hfill \mbox{\textit{Edexcel FD2 2024 Q2 [3]}}

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