10

10. [Maximum mark: 16]
    Consider the arithmetic sequence $a, p, q\dots$, where $a, p, q \neq 0$.

(a) Show that $2p - q = a$.

Consider the geometric sequence $a, s, t\dots$, where $a, s, t \neq 0$.

(b) Show that $s^2 = at$.

The first term of both sequences is $a$. It is given that $q = t = 1$.

(c) Show that $p > \frac{1}{2}$.

Consider the case where $a = 9, s > 0$ and $q = t = 1$.

(d) Write down the first four terms of the
(i) arithmetic sequence;

(ii) geometric sequence.

The arithmetic and the geometric sequence are used to form a new arithmetic sequence $u_n$. The first three terms of $u_n$ are $u_1 = 9 + \ln 9,\; u_2 = 5 + \ln 3$, and $u_3 = 1 + \ln 1$.

(e) (i) Find the common difference of the new sequence in terms of $\ln 3$.

(ii) Show that $\displaystyle \sum_{i=1}^{10} u_i = -90 - 25\ln 3$.