2. For ALL APPLICANTS.
For $k$ a positive integer, we define the polynomial $p _ { k } ( x )$ as
$$p _ { k } ( x ) = ( 1 + x ) \left( 1 + x ^ { 2 } \right) \left( 1 + x ^ { 3 } \right) \times \cdots \times \left( 1 + x ^ { k } \right) = a _ { 0 } + a _ { 1 } x + \cdots + a _ { N } x ^ { N } ,$$
denoting the coefficients of $p _ { k } ( x )$ as $a _ { 0 } , \ldots , a _ { N }$.
(i) Write down the degree $N$ of $p _ { k } ( x )$ in terms of $k$.
(ii) By setting $x = 1$, or otherwise, explain why
$$a _ { \max } \geqslant \frac { 2 ^ { k } } { N + 1 }$$
where $a _ { \text {max } }$ denotes the largest of the coefficients $a _ { 0 } , \ldots , a _ { N }$.
(iii) Fix $i \geqslant 0$. Explain why the value of $a _ { i }$ eventually becomes constant as $k$ increases.
A student correctly calculates for $k = 6$ that $p _ { 6 } ( x )$ equals
$$\begin{aligned}
& 1 + x + x ^ { 2 } + 2 x ^ { 3 } + 2 x ^ { 4 } + 3 x ^ { 5 } + 4 x ^ { 6 } + 4 x ^ { 7 } + 4 x ^ { 8 } + 5 x ^ { 9 } + 5 x ^ { 10 } + 5 x ^ { 11 } \\
& + 5 x ^ { 12 } + 4 x ^ { 13 } + 4 x ^ { 14 } + 4 x ^ { 15 } + 3 x ^ { 16 } + 2 x ^ { 17 } + 2 x ^ { 18 } + x ^ { 19 } + x ^ { 20 } + x ^ { 21 }
\end{aligned}$$
(iv) On the basis of this calculation, the student guesses that
$$a _ { i } = a _ { N - i } \quad \text { for } \quad 0 \leqslant i \leqslant N$$
By substituting $x ^ { - 1 }$ for $x$, or otherwise, show that the student's guess is correct for all positive integers $k$.
(v) On the basis of the same calculation, the student guesses that all whole numbers in the range $1,2 , \ldots , a _ { \text {max } }$ appear amongst the coefficients $a _ { 0 } , \ldots , a _ { N }$, for all positive integers $k$.
Use part (ii) to show that in this case the student's guess is wrong. Justify your answer.
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