Show by induction that
$$\forall N \in \mathbf { N } ^ { * } , \forall z \in D , \prod _ { k = 1 } ^ { N } \frac { 1 } { 1 - z ^ { k } } = \sum _ { n = 0 } ^ { + \infty } p _ { n , N } z ^ { n }$$

Show that for any non-zero natural integer $n$, $$D_{n} = n! \sum_{k=0}^{n} \frac{(-1)^{k}}{k!}$$

Let $c > 0$, and let $\left(a_n\right)_{n \in \mathbb{N}}$ be a sequence of positive real numbers such that $a_{n+1} \leq a_n - c(a_n)^2$ for all $n \in \mathbb{N}$. Show $a_n \leq a_0/(1 + nca_0)$ for all $n \in \mathbb{N}$. Hint: adapt the reasoning from question 10.c)

143- In proving the inequality $3^{n+1} > n!$ by mathematical induction, after finding a suitable number $m$, the inductive step relation for $k \geq m$ is which of the following?

(1) $k+1 > 2,\ m = 5$	(2) $k+1 > 2,\ m = 6$
(3) $(2k+1) > 4,\ m = 5$	(4) $(2k+1) > 4,\ m = 6$

143. We know that the sum of cubes of consecutive odd numbers starting from 1 equals the square of the sum of those numbers. What is the sum of cubes of consecutive odd numbers starting from 1 and ending at 19?
(1) $18800$ (2) $18900$ (3) $19800$ (4) $19900$

143. In the sequence $\{U_n\}$, with initial conditions $U_1 = U_2 = 1$ and recurrence $U_{n+1} = U_n + U_{n-1}$, using induction, the expression $(U_n^2 - U_{n+1} \times U_{n-1})$ equals which number?
(1) $-1$ (2) $1$ (3) $(-1)^n$ (4) $(-1)^{n+1}$

Let $\left\{ u _ { n } \right\} _ { n \geq 1 }$ be a sequence of real numbers defined as $u _ { 1 } = 1$ and
$$u _ { n + 1 } = u _ { n } + \frac { 1 } { u _ { n } } \text { for all } n \geq 1 .$$
Prove that $u _ { n } \leq \frac { 3 \sqrt { n } } { 2 }$ for all $n$.

3. Let n be any positive integer. Prove that:
$$\sum _ { k = 0 } ^ { m } \frac { \binom { 2 n - k } { k } } { \binom { 2 n - k } { n } } \cdot \frac { ( 2 n - 4 k + 1 ) } { ( 2 n - 2 k + 1 ) } 2 ^ { n - 2 k } = \frac { \binom { n } { m } } { \binom { 2 n - 2 m } { n - m } } 2 ^ { n - 2 m }$$
for each nonnegative integer $\mathrm { m } \leq \mathrm { n }$. (Here $\left. \binom { p } { q } = \square ^ { p } C _ { q } \right)$

9. (a) A coin has probability p of showing head when tossed. It is tossed n times. Let pn , denote the probability that no two (or more) consecutive heads occur. Prove that $\mathrm { p } 1 = 1 , \mathrm { p } ^ { 2 } = 1 - \mathrm { p } _ { 2 }$ and $\mathrm { p } = ( 1 -$ p ). $\mathrm { pn } - 1 + \mathrm { p } ( 1 - \mathrm { p } ) \mathrm { pn } - 2$ for all $\mathrm { n } \geq 3$.
(b) In (a), prove by induction on n , that $\mathrm { pn } = \mathrm { A } \alpha \mathrm { n } + \mathrm { B } \beta \mathrm { n }$ for all $\mathrm { n } \geq 1$, where $\alpha$ and $\beta$ are the roots of the quadratic x2-(1-p) $x - p ( 1 - p ) = 0$ and
$$A = \frac { p ^ { 2 } + \beta - 1 } { \alpha \beta - \alpha ^ { 2 } } , B = \frac { p ^ { 2 } + \alpha - 1 } { \alpha \beta - \beta ^ { 2 } } .$$

1. Let ABC and PQR be any two triangles in the same plane. Assume that the perpendiculars from the points $\mathrm { A } , \mathrm { B } , \mathrm { C }$ to the sides $\mathrm { QR } , \mathrm { RP } , \mathrm { PQ }$ respectively are concurrent. Using vector methods or otherwise, prove that the perpendiculars from $\mathrm { P } , \mathrm { Q } , \mathrm { R }$ to $\mathrm { BC } , \mathrm { CA } , \mathrm { AB }$ respectively are also concurrent.

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a thrice differentiable function such that $f ( 0 ) = 0 , f ( 1 ) = 1 , f ( 2 ) = - 1 , f ( 3 ) = 2$ and $f ( 4 ) = - 2$. Then, the minimum number of zeros of $\left( 3 f ^ { \prime } f ^ { \prime \prime } + f f ^ { \prime \prime \prime } \right) ( x )$ is $\_\_\_\_$

A third-degree polynomial $\mathbf { P } ( \mathbf { x } )$ with leading coefficient 1 and its derivative $P ^ { \prime } ( x )$ satisfy
$$P ( 0 ) = P ( 1 ) = P ^ { \prime } ( 1 ) = 0$$
Accordingly, what is the value of $P ( - 1 )$?
A) 3
B) 1
C) 0
D) - 2
E) - 4