 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2024

Paper 2

The <floor-half generating function> is
$$
H(x)=\sum_{n\geq1}\left\lfloor\frac n2\right\rfloor x^n
=\frac{x^2}{(1-x)^2(1+x)}.
$$
Summing $u_n-2u_{n-1}=\lfloor n/2\rfloor$ for $n\geq1$ gives
$$
G(x)-1-2xG(x)=H(x),
$$
so
$$
\boxed{
G(x)=\frac{1+\dfrac{x^2}{(1-x)^2(1+x)}}{1-2x}}.
$$

Since
$$
\left\lfloor\frac n2\right\rfloor
=\frac n2-\frac14+\frac14(-1)^n,
$$
a particular solution is
$$
-\frac n2-\frac34+\frac1{12}(-1)^n.
$$
Adding the homogeneous term $C2^n$ and imposing $u_0=1$ gives $C=5/3$. Therefore
$$
\boxed{
u_n=\frac53\,2^n-\frac n2-\frac34+\frac1{12}(-1)^n}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 2 / 6A / b / Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 2 / 6A / b / Solution

 Ancestors (11)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 2 / 6A / b / Solution

 Ancestors (11)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)