Finding unknown power and constant

6. Given that the binomial expansion, in ascending powers of \(x\), of $$\frac { 6 } { \sqrt { } \left( 9 + A x ^ { 2 } \right) } , \quad | x | < \frac { 3 } { \sqrt { } | A | }$$ is \(\quad B - \frac { 2 } { 3 } x ^ { 2 } + C x ^ { 4 } + \ldots\)

1. find the values of the constants \(A , B\) and \(C\).
2. Hence find the coefficient of \(x ^ { 6 }\)

2. \(\quad \mathrm { f } ( x ) = ( 2 + k x ) ^ { - 3 } , \quad | k x | < 2\), where \(k\) is a positive constant The binomial expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\) is $$A + B x + \frac { 243 } { 16 } x ^ { 2 }$$ where \(A\) and \(B\) are constants.

1. Write down the value of \(A\).
2. Find the value of \(k\).
3. Find the value of \(B\).

1. Given that \(k\) is a constant and the binomial expansion of

$$\sqrt { 1 + k x } \quad | k x | < 1$$ in ascending powers of \(x\) up to the term in \(x ^ { 3 }\) is $$1 + \frac { 1 } { 8 } x + A x ^ { 2 } + B x ^ { 3 }$$

1. 1. find the value of \(k\),
   2. find the value of the constant \(A\) and the constant \(B\).
2. Use the expansion to find an approximate value to \(\sqrt { 1.15 }\) Show your working and give your answer to 6 decimal places.

8. Given that for small values of \(x\) $$( 1 + a x ) ^ { n } \approx 1 - 24 x + 270 x ^ { 2 }$$ where \(n\) is an integer and \(n > 1\),

1. show that \(n = 16\) and find the value of \(a\),
2. use your value of \(a\) and a suitable value of \(x\) to estimate the value of (0.9985) \({ } ^ { 16 }\), giving your answer to 5 decimal places.

2 Given that \(\left( 1 + \frac { x } { p } \right) ^ { q } = 1 - x + \frac { 3 } { 4 } x ^ { 2 } + \ldots\), find \(p\) and \(q\), and state the set of values of \(x\) for which the expansion is valid.

3. The first four terms in the series expansion of \(( 1 + a x ) ^ { n }\) in ascending powers of \(x\) are $$1 - 4 x + 24 x ^ { 2 } + k x ^ { 3 }$$ where \(a , n\) and \(k\) are constants and \(| a x | < 1\).

1. Find the values of \(a\) and \(n\).
2. Show that \(k = - 160\).

2 Given the binomial expansion \(( 1 + q x ) ^ { p } = 1 - x + 2 x ^ { 2 } + \ldots\), find the values of \(p\) and \(q\). Hence state the set of values of \(x\) for which the expansion is valid. [6]

1.In the binomial expansion of $$\left( 1 + \frac { 12 n } { 5 } x \right) ^ { n }$$ the coefficients of \(x ^ { 2 }\) and \(x ^ { 3 }\) are equal and non-zero.

1. Find the possible values of \(n\) .
   (4)
2. State,giving a reason,which value of \(n\) gives a valid expansion when \(x = \frac { 1 } { 2 }\) (2)

7 Given that the binomial expansion of \(( 1 + k x ) ^ { n }\) is \(1 - 6 x + 30 x ^ { 2 } + \ldots\), find the values of \(n\) and \(k\). State the set of values of \(x\) for which this expansion is valid.

3 The function \(\mathrm { f } ( x )\) is given by \(\mathrm { f } ( x ) = ( 1 - a x ) ^ { - 3 }\), where \(a\) is a non-zero constant. In the binomial expansion of \(\mathrm { f } ( x )\), the coefficients of \(x\) and \(x ^ { 2 }\) are equal.

1. Find the value of \(a\).
2. Using this value for \(a\),
   1. state the set of values of \(x\) for which the binomial expansion is valid,
   2. write down the quadratic function which approximates \(\mathrm { f } ( x )\) when \(x\) is small.

7. Given that for small values of \(x\) $$( 1 + a x ) ^ { n } \approx 1 - 24 x + 270 x ^ { 2 } ,$$ where \(n\) is an integer and \(n > 1\),

1. show that \(n = 16\) and find the value of \(a\),
2. use your value of \(a\) and a suitable value of \(x\) to estimate the value of (0.9985) \({ } ^ { 16 }\), giving your answer to 5 decimal places.

1. The first four terms in the series expansion of \(( 1 + a x ) ^ { n }\) in ascending powers of \(x\) are

$$1 - 4 x + 24 x ^ { 2 } + k x ^ { 3 }$$ where \(a , n\) and \(k\) are constants and \(| a x | < 1\).

1. Find the values of \(a\) and \(n\).
2. Show that \(k = - 160\).
   3. continued

Given the binomial expansion \((1 + qx)^p = 1 - x + 2x^2 + \ldots\), find the values of \(p\) and \(q\). Hence state the set of values of \(x\) for which the expansion is valid. [6]

In the binomial expansion of $$(1 - 4x)^p, \quad |x| < \frac{1}{4},$$ the coefficient of \(x^2\) is equal to the coefficient of \(x^4\) and the coefficient of \(x^3\) is positive. Find the value of \(p\). [9]