Standard +0.8 This is a first-order linear non-homogeneous recurrence relation requiring the complementary function plus particular integral method. While the technique is standard for FD2, it demands careful algebraic manipulation to find the particular solution form (trying u_n = k·2^n), then applying the initial condition u_1 = u_2 requires solving for the constant. This is moderately challenging as it tests both procedural fluency and algebraic accuracy across multiple steps, placing it above average difficulty.
2. (a) Find the general solution of the recurrence relation
$$u _ { n + 1 } = 3 u _ { n } + 2 ^ { n } \quad n \geqslant 1$$
(b) Find the particular solution of this recurrence relation for which \(u _ { 1 } = u _ { 2 }\)
Using condition \(u_1 = u_2\) to find \(A\ (= \tfrac{1}{3})\); dependent on general solution being of form \(\pm\lambda(3)^n \pm \mu(2)^n\)
\(u_n = 3^{n-1} - 2^n\)
A1
CAO; must include \(u_n = \ldots\); if neither general nor particular solution given in terms of \(u_n\), award if correct expression in terms of \(n\) seen
# Question 2:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Auxiliary equation $m - 3 = 0 \Rightarrow$ complementary function is $A(3)^n$ | B1 | CAO for complementary function |
| Consider trial solution $u_n = k(2^n)$, so $2k(2^n) = 3k(2^n) + 2^n$ | M1 | Substituting correct trial solution into recurrence relation; allow substitution of $u_n = k(2^n)$ into $u_n = 3u_{n-1} + 2^{n-1}$ but not $u_n = 3u_{n-1} + 2^n$ |
| $k = -1$ | A1 | CAO |
| General solution is $u_n = A(3)^n - 2^n$ | A1 | CAO for general solution — must include $u_n = \ldots$ |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $u_1 = u_2 \Rightarrow 3A - 2 = 9A - 4 \Rightarrow A = \ldots$ | M1 | Using condition $u_1 = u_2$ to find $A\ (= \tfrac{1}{3})$; dependent on general solution being of form $\pm\lambda(3)^n \pm \mu(2)^n$ |
| $u_n = 3^{n-1} - 2^n$ | A1 | CAO; must include $u_n = \ldots$; if neither general nor particular solution given in terms of $u_n$, award if correct expression in terms of $n$ seen |
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2. (a) Find the general solution of the recurrence relation
$$u _ { n + 1 } = 3 u _ { n } + 2 ^ { n } \quad n \geqslant 1$$
(b) Find the particular solution of this recurrence relation for which $u _ { 1 } = u _ { 2 }$\\
\hfill \mbox{\textit{Edexcel FD2 AS 2019 Q2 [6]}}