Arithmetic/geometric progression coefficients

4 When \(( 1 + a x ) ^ { - 2 }\), where \(a\) is a positive constant, is expanded in ascending powers of \(x\), the coefficients of \(x\) and \(x ^ { 3 }\) are equal.

1. Find the exact value of \(a\).
2. When \(a\) has this value, obtain the expansion up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.

6．Given that the coefficients of \(x , x ^ { 2 }\) and \(x ^ { 4 }\) in the expansion of \(( 1 + k x ) ^ { n }\) ，where \(n \geq 4\) and \(k\) is a positive constant，are the consecutive terms of a geometric series，
（a）show that \(k = \frac { 6 ( n - 1 ) } { ( n - 2 ) ( n - 3 ) }\) ．
（b）Given further that both \(n\) and \(k\) are positive integers，find all possible pairs of values for \(n\) and \(k\) ．You should show clearly how you know that you have found all possible pairs of values．
（c）For the case where \(k = 1.4\) ，find the value of the positive integer \(n\) ．
（d）Given that \(k = 1.4 , n\) is a positive integer and that the first term of the geometric series is the coefficient of \(x\) ，estimate how many terms are required for the sum of the geometric series to exceed \(1.12 \times 10 ^ { 12 }\) ．［You may assume that \(\log _ { 10 } 4 \approx 0.6\) and \(\log _ { 10 } 5 \approx 0.7\) ．］

1．In the binomial expansion of $$( 1 - 8 x ) ^ { p } \quad | x | < \frac { 1 } { 8 }$$ where \(p\) is a positive constant，
－the sum of the coefficient of \(x\) and the coefficient of \(x ^ { 2 }\) is equal to the coefficient of \(x ^ { 3 }\)
－the coefficient of \(x ^ { 2 }\) is positive
Determine the value of \(p\) ．
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