A sequence \(x_r\) for \(r = 0, 1, 2, 3, \ldots\) is such that \(x_r = x_{r-1} + \ln(r + 1) + 3, \qquad \text{for all}\hspace{0.5em} r \geq 1.\) It is given that \(x_0 = 6\).
Find the exact values of \(x_1, x_2\) and \(x_3\), leaving your answers in the form \(A + \ln(B)\), where \(A\) and \(B\) are constants to be determined.[2]
Find \(x_n\) in terms of \(n\).[2]
Arithmetic progression \(U\) has first term \(a\) and common difference \(u\). Another arithmetic progression \(V\) has first term \(b\) and common difference \(v\). It is given that
\(a\) is a non-zero integer,
the product of the first 3 terms of \(U\) is 0,
the sum of the first terms and common differences of both progressions is \(-11\),
the \(12^{\text{th}}\) term of \(U\) is three times of the \(10^{\text{th}}\) term of \(V\),
the sum of first 50 even-numbered terms of \(V\) is 5800.
Write down and solve equations to find the values of \(a, u, b\) and \(v\).[4]
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