Edexcel FP2 2022 June — Question 6 6 marks

Exam BoardEdexcel
ModuleFP2 (Further Pure Mathematics 2)
Year2022
SessionJune
Marks6
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Mark schemeDownload PDF ↗
TopicSequences and series, recurrence and convergence
TypeRecurrence relation solving for closed form
DifficultyChallenging +1.2 This is a second-order linear recurrence relation with a non-homogeneous term (2^n). While it requires knowledge of characteristic equations, particular solutions, and applying initial conditions, it follows a standard FP2 template with straightforward algebra. The repeated root (r=1) adds mild complexity, but the overall solution path is well-established and methodical rather than requiring novel insight.
Spec4.10e Second order non-homogeneous: complementary + particular integral
6. (a) Determine the general solution of the recurrence relation $$u _ { n } = 2 u _ { n - 1 } - u _ { n - 2 } + 2 ^ { n } \quad n \geqslant 2$$ (b) Hence solve this recurrence relation given that \(u _ { 0 } = 2 u _ { 1 }\) and \(u _ { 4 } = 3 u _ { 2 }\)
Question 6:
Part (a):
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(l^2 - 2l + 1 = 0 \Rightarrow l = \ldots \{\lambda = 1\}\)M1 Forms and solves auxiliary equation
\(u_n = A + Bn\)A1 Correct complementary function
\(u_n = \mu(2)^n \Rightarrow \mu(2)^n = 2\mu(2)^{n-1} - \mu(2)^{n-2} + (2)^n\); \(\Rightarrow 4\mu(2)^{n-2} = 4\mu(2)^{n-2} - \mu(2)^{n-2} + 4(2)^{n-2}\); \(\Rightarrow \mu = \ldots\{4\}\)M1 Correct form for particular solution; substitutes into recurrence relation to find PS
\(u_n = A + Bn + 4(2)^n\) or \(u_n = A + Bn + (2)^{n+2}\)A1 Correct general solution
Part (b):
AnswerMarks Guidance
Answer/WorkingMark Guidance
Uses \(u_0 = 2u_1\) and \(u_4 = 3u_2\) (or equivalent) to form two equations in \(A\) and \(B\); solves simultaneouslyM1 Must have two constants to score any marks; complete method using given information
\(u_n = 28 - 20n + 4(2)^n\) or \(u_n = 28 - 20n + (2)^{n+2}\)A1 
## Question 6:

### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $l^2 - 2l + 1 = 0 \Rightarrow l = \ldots \{\lambda = 1\}$ | M1 | Forms and solves auxiliary equation |
| $u_n = A + Bn$ | A1 | Correct complementary function |
| $u_n = \mu(2)^n \Rightarrow \mu(2)^n = 2\mu(2)^{n-1} - \mu(2)^{n-2} + (2)^n$; $\Rightarrow 4\mu(2)^{n-2} = 4\mu(2)^{n-2} - \mu(2)^{n-2} + 4(2)^{n-2}$; $\Rightarrow \mu = \ldots\{4\}$ | M1 | Correct form for particular solution; substitutes into recurrence relation to find PS |
| $u_n = A + Bn + 4(2)^n$ or $u_n = A + Bn + (2)^{n+2}$ | A1 | Correct general solution |

### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Uses $u_0 = 2u_1$ and $u_4 = 3u_2$ (or equivalent) to form two equations in $A$ and $B$; solves simultaneously | M1 | Must have two constants to score any marks; complete method using given information |
| $u_n = 28 - 20n + 4(2)^n$ or $u_n = 28 - 20n + (2)^{n+2}$ | A1 | |

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6. (a) Determine the general solution of the recurrence relation
$$u _ { n } = 2 u _ { n - 1 } - u _ { n - 2 } + 2 ^ { n } \quad n \geqslant 2$$
(b) Hence solve this recurrence relation given that $u _ { 0 } = 2 u _ { 1 }$ and $u _ { 4 } = 3 u _ { 2 }$

\hfill \mbox{\textit{Edexcel FP2 2022 Q6 [6]}}
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