Finding unknown constant from coefficient

5 When \(( 1 + 2 x ) ( 1 + a x ) ^ { \frac { 2 } { 3 } }\), where \(a\) is a constant, is expanded in ascending powers of \(x\), the coefficient of the term in \(x\) is zero.

1. Find the value of \(a\).
2. When \(a\) has this value, find the term in \(x ^ { 3 }\) in the expansion of \(( 1 + 2 x ) ( 1 + a x ) ^ { \frac { 2 } { 3 } }\), simplifying the coefficient.

4. (a) Find the binomial expansion of $$( 1 + p x ) ^ { - 4 } , \quad | p x | < 1$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), giving each coefficient as simply as possible in terms of the constant \(p\). $$f ( x ) = \frac { 3 + 4 x } { ( 1 + p x ) ^ { 4 } } \quad | p x | < 1$$ where \(p\) is a positive constant. In the series expansion of \(\mathrm { f } ( x )\), the coefficient of \(x ^ { 2 }\) is twice the coefficient of \(x\).
(b) Find the value of \(p\).
(c) Hence find the coefficient of \(x ^ { 3 }\) in the series expansion of \(\mathrm { f } ( x )\), giving your answer as a simplified fraction.

1. (a) Use the binomial series to expand

$$\frac { 1 } { ( 2 - 3 x ) ^ { 3 } } \quad | x | < \frac { 2 } { 3 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), giving each term as a simplified fraction. $$f ( x ) = \frac { 4 + k x } { ( 2 - 3 x ) ^ { 3 } } \quad \text { where } k \text { is a constant and } | x | < \frac { 2 } { 3 }$$ Given that the series expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), is $$\frac { 1 } { 2 } + A x + \frac { 81 } { 16 } x ^ { 2 } + \cdots$$ where \(A\) is a constant,
(b) find the value of \(k\),
(c) find the value of \(A\).

1. (a) Use the binomial expansion to expand

$$( 4 - 5 x ) ^ { - \frac { 1 } { 2 } } \quad | x | < \frac { 4 } { 5 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\) giving each coefficient as a fully simplified fraction. $$f ( x ) = \frac { 2 + k x } { \sqrt { 4 - 5 x } } \quad \text { where } k \text { is a constant and } | x | < \frac { 4 } { 5 }$$ Given that the series expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), is $$1 + \frac { 3 } { 10 } x + m x ^ { 2 } + \ldots \quad \text { where } m \text { is a constant }$$ (b) find the value of \(k\),
(c) find the value of \(m\).

1. (i) Expand \(( 1 + a x ) ^ { - 3 } , | a x | < 1\), in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\). Give each coefficient as simply as possible in terms of the constant \(a\).

Given that the coefficient of \(x ^ { 2 }\) in the expansion of \(\frac { 6 - x } { ( 1 + a x ) ^ { 3 } } , | a x | < 1\), is 3 ,
(ii) find the two possible values of \(a\). Given also that \(a < 0\),
(iii) show that the coefficient of \(x ^ { 3 }\) in the expansion of \(\frac { 6 - x } { ( 1 + a x ) ^ { 3 } }\) is \(\frac { 14 } { 9 }\).

3. (a) Find the binomial expansion of $$( 1 + a x ) ^ { - 3 } , \quad | a x | < 1$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), giving each coefficient as simply as possible in terms of the constant \(a\). $$f ( x ) = \frac { 2 + 3 x } { ( 1 + a x ) ^ { 3 } } , \quad | a x | < 1$$ In the series expansion of \(\mathrm { f } ( x )\), the coefficient of \(x ^ { 2 }\) is 3
Given that \(a < 0\)
(b) find the value of the constant \(a\),
(c) find the coefficient of \(x ^ { 3 }\) in the series expansion of \(\mathrm { f } ( x )\), giving your answer as a simplified fraction.

8

1. Find the first three terms in the expansion of \(( 4 - x ) ^ { - \frac { 1 } { 2 } }\) in ascending powers of \(x\).
2. The expansion of \(\frac { a + b x } { \sqrt { 4 - x } }\) is \(16 - x \ldots\). Find the values of the constants \(a\) and \(b\).

8

1. Find the first three terms in the expansion of \(( 4 + 3 x ) ^ { \frac { 3 } { 2 } }\) in ascending powers of \(x\).
2. State the range of values of \(x\) for which the expansion in part (a) is valid.
3. In the expansion of \(( 4 + 3 x ) ^ { \frac { 3 } { 2 } } ( 1 + a x ) ^ { 2 }\) the coefficient of \(x ^ { 2 }\) is \(\frac { 107 } { 16 }\). Determine the possible values of the constant \(a\).

4. (a) Expand \(( 1 + a x ) ^ { - 3 } , | a x | < 1\), in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\). Give each coefficient as simply as possible in terms of the constant \(a\). Given that the coefficient of \(x ^ { 2 }\) in the expansion of \(\frac { 6 - x } { ( 1 + a x ) ^ { 3 } } , | a x | < 1\), is 3 ,
(b) find the two possible values of \(a\). Given also that \(a < 0\),
(c) show that the coefficient of \(x ^ { 3 }\) in the expansion of \(\frac { 6 - x } { ( 1 + a x ) ^ { 3 } }\) is \(\frac { 14 } { 9 }\).
4. continued

3. (a) Find and simplify the first three terms in the expansion, in ascending powers of \(x\), of \(\left( 2 + \frac { 1 } { 3 } k x \right) ^ { 6 }\), where \(k\) is a constant.
(b) In the expansion of \(( 3 - 4 x ) \left( 2 + \frac { 1 } { 3 } k x \right) ^ { 6 }\), the constant term is equal to the coefficient of \(x ^ { 2 }\). Determine the exact value of \(k\), given that \(k\) is positive.
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