| Exam Board | Edexcel |
|---|---|
| Module | FD2 (Further Decision 2) |
| Year | 2024 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Recurrence relation solving for closed form |
| Difficulty | Challenging +1.2 This is a second-order linear non-homogeneous recurrence relation requiring the characteristic equation method plus finding a particular solution for the polynomial term. While it involves multiple techniques (solving quadratic, finding particular solution with linear form, applying initial conditions), these are standard Further Maths procedures with no novel insight required. The constraint that all terms are positive adds a minor twist to part (b) but doesn't significantly increase difficulty. Harder than typical A-level Core questions but routine for FP2/FD2 students. |
| Spec | 4.04e Line intersections: parallel, skew, or intersecting |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Auxiliary equation: \(2m^2 + 5m - 3 = 0 \Rightarrow m = \ldots\) | M1 | |
| Complementary function: \(A(0.5)^n + B(-3)^n\) | A1 | |
| Particular solution: try \(u_n = an + b\), substitute into recurrence relation | M1 | |
| \(2(a(n+2)+b) + 5(a(n+1)+b) = 3(an+b) + 8n + 2\); comparing gives \(4a = 8\), \(9a + 4b = 2\) | dM1 | Dependent on previous M1 |
| \(u_n = A(0.5)^n + B(-3)^n + 2n - 4\) | A1ft | Follow through on CF |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(A + B - 4 = 1\) and \(0.5A - 3B - 2 = k\), leading to values of \(A\) and \(B\) | ddM1 | Dependent on both previous M marks |
| \(u_n = \left(\frac{34+2k}{7}\right)(0.5)^n + \left(\frac{1-2k}{7}\right)(-3)^n + 2n - 4\); setting \(\frac{1-2k}{7} = 0\) | dddM1 | Dependent on all three previous M marks |
| \(k = 0.5\) | A1 | CAO |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Correct auxiliary equation with attempt to solve, leading to two distinct values of \(m\) | M1 | — |
| Correct complementary function | A1 | Accept if seen in subsequent working |
| Correct form for particular solution; substitute \(n+2\), \(n+1\), \(n\) into recurrence relation | M1 | — |
| Compare coefficients; set up one equation in \(a\) only and one equation in \(a\) and \(b\); giving \(a=2\) and \(b=-4\) | dM1 | Dependent on previous M mark |
| Correct general solution: C.F. \(+ 2n - 4\) | A1ft | Must use their C.F.; award for fully correct solution seen here or used in (b) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use correct initial conditions to form simultaneous equations in \(A\), \(B\) and \(k\); attempt to solve for \(A\) and \(B\) | ddM1 | Dependent on previous two M marks; award for either: eliminating \(A\) or \(B\) giving \(7A = 2k+34\) or \(7B = 1-2k\); or eliminating both \(A\) and \(B\) giving \(A = \dfrac{34+2k}{7},\ B = \dfrac{1-2k}{7}\) |
| Set coefficient (linear expression in \(k\)) of \((-3)^n\) equal to zero | dddM1 | Dependent on previous three M marks; requires at least one root of auxiliary equation to be negative |
| Correct value of \(k\) (CAO, from correct working) | A1 | — |
## Question 8:
**(a)**
| Answer/Working | Mark | Guidance |
|---|---|---|
| Auxiliary equation: $2m^2 + 5m - 3 = 0 \Rightarrow m = \ldots$ | M1 | |
| Complementary function: $A(0.5)^n + B(-3)^n$ | A1 | |
| Particular solution: try $u_n = an + b$, substitute into recurrence relation | M1 | |
| $2(a(n+2)+b) + 5(a(n+1)+b) = 3(an+b) + 8n + 2$; comparing gives $4a = 8$, $9a + 4b = 2$ | dM1 | Dependent on previous M1 |
| $u_n = A(0.5)^n + B(-3)^n + 2n - 4$ | A1ft | Follow through on CF |
**(b)**
| Answer/Working | Mark | Guidance |
|---|---|---|
| $A + B - 4 = 1$ and $0.5A - 3B - 2 = k$, leading to values of $A$ and $B$ | ddM1 | Dependent on both previous M marks |
| $u_n = \left(\frac{34+2k}{7}\right)(0.5)^n + \left(\frac{1-2k}{7}\right)(-3)^n + 2n - 4$; setting $\frac{1-2k}{7} = 0$ | dddM1 | Dependent on all three previous M marks |
| $k = 0.5$ | A1 | CAO |
## Question 8:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Correct auxiliary equation with attempt to solve, leading to two distinct values of $m$ | M1 | — |
| Correct complementary function | A1 | Accept if seen in subsequent working |
| Correct form for particular solution; substitute $n+2$, $n+1$, $n$ into recurrence relation | M1 | — |
| Compare coefficients; set up one equation in $a$ only and one equation in $a$ and $b$; giving $a=2$ and $b=-4$ | dM1 | Dependent on previous M mark |
| Correct general solution: C.F. $+ 2n - 4$ | A1ft | Must use their C.F.; award for fully correct solution seen here or used in (b) |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use correct initial conditions to form simultaneous equations in $A$, $B$ and $k$; attempt to solve for $A$ and $B$ | ddM1 | Dependent on previous two M marks; award for **either**: eliminating $A$ **or** $B$ giving $7A = 2k+34$ or $7B = 1-2k$; **or** eliminating both $A$ and $B$ giving $A = \dfrac{34+2k}{7},\ B = \dfrac{1-2k}{7}$ |
| Set coefficient (linear expression in $k$) of $(-3)^n$ equal to zero | dddM1 | Dependent on previous three M marks; requires at least one root of auxiliary equation to be negative |
| Correct value of $k$ (CAO, from correct working) | A1 | — |8. A sequence $\left\{ u _ { n } \right\}$, where $n \geqslant 0$, satisfies the recurrence relation
$$2 u _ { n + 2 } + 5 u _ { n + 1 } = 3 u _ { n } + 8 n + 2$$
\begin{enumerate}[label=(\alph*)]
\item Find the general solution of this recurrence relation.\\
(5)
A particular solution of this recurrence relation has $u _ { 0 } = 1$ and $u _ { 1 } = k$, where $k$ is a positive constant. All terms of the sequence are positive.
\item Determine the value of $k$.\\
(3)
\end{enumerate}
\hfill \mbox{\textit{Edexcel FD2 2024 Q8 [8]}}