| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2020 |
| Session | June |
| Marks | 9 |
| 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 recurrence relation with a non-homogeneous term (3n). While it requires knowledge of the auxiliary equation method, finding particular integrals, and combining solutions—techniques beyond basic A-level—it's a standard FP2 exercise with straightforward arithmetic. The auxiliary equation (3λ-2)²=0 gives a repeated root requiring the xα^n form, and the particular integral is a linear function. More routine than typical FP2 proof questions but still requires multiple Further Maths techniques. |
| Spec | 1.04e Sequences: nth term and recurrence relations4.06a Summation formulae: sum of r, r^2, r^3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Auxiliary equation is \(9r^2 - 12r + 4 = 0\), so \(r = \ldots\) | M1 | Forms and solves the auxiliary equation |
| \((3r-2)^2 = 0 \Rightarrow r = \dfrac{2}{3}\) is repeated root | A1 | Correct (repeated) root found |
| Complementary function is \(x_n = (A+Bn)\left(\dfrac{2}{3}\right)^n\) or \(A\left(\dfrac{2}{3}\right)^n + Bn\left(\dfrac{2}{3}\right)^n\) | M1 | Forms correct complementary function for their real root(s); \((A+Bn)r^n\) if repeated root, or \(Ar_1^n + Br_2^n\) if distinct real roots |
| Try particular solution \(y_n = an+b \Rightarrow 9(a(n+2)+b) - 12(a(n+1)+b) + 4(an+b) = 3n\) | M1 | Attempts particular solution of correct form (\(an+b\) or higher order polynomial) |
| \(\Rightarrow an + 6a + b = 3n \Rightarrow a = \ldots, b = \ldots\) | dM1 | Expands and solves for \(a\) and \(b\) |
| \(a = 3,\ b = -18\) | A1 | Correct values for \(a\) and \(b\) |
| General solution is \(u_n = x_n + y_n = (A+Bn)\left(\dfrac{2}{3}\right)^n + 3n - 18\) | B1ft | Forms general solution as sum of complementary function and particular solution, with correct \(a\) and \(b\) |
| \(u_1 = 1 \Rightarrow 1 = \left(\dfrac{2}{3}\right)(A+B) - 15\)<br>\(u_2 = 4 \Rightarrow 4 = \left(\dfrac{4}{9}\right)(A+2B) - 12\) \(\bigg\} A = \ldots, B = \ldots\) | M1 | Applies initial values and solves for the constants |
| \(u_n = 12(n+1)\left(\dfrac{2}{3}\right)^n + 3n - 18\) oe | A1 | Correct answer |
## Question 2:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Auxiliary equation is $9r^2 - 12r + 4 = 0$, so $r = \ldots$ | M1 | Forms and solves the auxiliary equation |
| $(3r-2)^2 = 0 \Rightarrow r = \dfrac{2}{3}$ is repeated root | A1 | Correct (repeated) root found |
| Complementary function is $x_n = (A+Bn)\left(\dfrac{2}{3}\right)^n$ or $A\left(\dfrac{2}{3}\right)^n + Bn\left(\dfrac{2}{3}\right)^n$ | M1 | Forms correct complementary function for their real root(s); $(A+Bn)r^n$ if repeated root, or $Ar_1^n + Br_2^n$ if distinct real roots |
| Try particular solution $y_n = an+b \Rightarrow 9(a(n+2)+b) - 12(a(n+1)+b) + 4(an+b) = 3n$ | M1 | Attempts particular solution of correct form ($an+b$ or higher order polynomial) |
| $\Rightarrow an + 6a + b = 3n \Rightarrow a = \ldots, b = \ldots$ | dM1 | Expands and solves for $a$ and $b$ |
| $a = 3,\ b = -18$ | A1 | Correct values for $a$ and $b$ |
| General solution is $u_n = x_n + y_n = (A+Bn)\left(\dfrac{2}{3}\right)^n + 3n - 18$ | B1ft | Forms general solution as sum of complementary function and particular solution, with correct $a$ and $b$ |
| $u_1 = 1 \Rightarrow 1 = \left(\dfrac{2}{3}\right)(A+B) - 15$<br>$u_2 = 4 \Rightarrow 4 = \left(\dfrac{4}{9}\right)(A+2B) - 12$ $\bigg\} A = \ldots, B = \ldots$ | M1 | Applies initial values and solves for the constants |
| $u_n = 12(n+1)\left(\dfrac{2}{3}\right)^n + 3n - 18$ oe | A1 | Correct answer |
**(9 marks)**\begin{enumerate}
\item Solve the recurrence system
\end{enumerate}
$$\begin{gathered}
u _ { 1 } = 1 \quad u _ { 2 } = 4 \\
9 u _ { n + 2 } - 12 u _ { n + 1 } + 4 u _ { n } = 3 n
\end{gathered}$$
\hfill \mbox{\textit{Edexcel FP2 2020 Q2 [9]}}