First few terms given find parameters

4 The first three terms in the expansion of \(( 2 + a x ) ^ { n }\), in ascending powers of \(x\), are \(32 - 40 x + b x ^ { 2 }\). Find the values of the constants \(n , a\) and \(b\).

7. (a) Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of \(( 1 + k x ) ^ { 8 }\), where \(k\) is a non-zero constant. Give each term in its simplest form. Given that the coefficient of \(x ^ { 3 }\) in this expansion is 1512
(b) find the value of \(k\).

10. The first 3 terms, in ascending powers of \(x\), in the binomial expansion of \(( 1 + a x ) ^ { 20 }\) are given by $$1 + 4 x + p x ^ { 2 }$$ where \(a\) and \(p\) are constants.

1. Find the value of \(a\).
2. Find the value of \(p\). One of the terms in the binomial expansion of \(( 1 + a x ) ^ { 20 }\) is \(q x ^ { 4 }\), where \(q\) is a constant.
3. Find the value of \(q\).

15. The binomial expansion, in ascending powers of \(x\), of \(( 1 + k x ) ^ { n }\) is $$1 + 36 x + 126 k x ^ { 2 } + \ldots$$ where \(k\) is a non-zero constant and \(n\) is a positive integer.

1. Show that \(n k ( n - 1 ) = 252\)
2. Find the value of \(k\) and the value of \(n\).
3. Using the values of \(k\) and \(n\), find the coefficient of \(x ^ { 3 }\) in the binomial expansion of \(( 1 + k x ) ^ { n }\)
   

8. Given that $$1 + 12 x + 70 x ^ { 2 } + \ldots$$ is the binomial expansion, in ascending powers of \(x\) of \(( 1 + b x ) ^ { n }\), where \(n \in \mathbb { N }\) and \(b\) is a constant,

1. show that \(n b = 12\)
2. find the values of the constants \(b\) and \(n\).

1. The first three terms in ascending powers of \(x\) in the binomial expansion of \(( 1 + p x ) ^ { 8 }\) are given by

$$1 + 12 x + q x ^ { 2 }$$ where \(p\) and \(q\) are constants.
Find the value of \(p\) and the value of \(q\).

1. The binomial expansion, in ascending powers of \(x\), of

$$( 3 + p x ) ^ { 5 }$$ where \(p\) is a constant, can be written in the form $$A + B x + C x ^ { 2 } + D x ^ { 3 } \ldots$$ where \(A\), \(B\), \(C\) and \(D\) are constants.

1. Find the value of \(A\) Given that
   • \(B = 18 D\)
   • \(p < 0\)
   • find
     1. the value of \(p\)
     2. the value of \(C\)

1. The first three terms, in ascending powers of \(x\), of the binomial expansion of \(( 1 + k x ) ^ { 16 }\) are

$$1 , - 4 x \text { and } p x ^ { 2 }$$ where \(k\) and \(p\) are constants.

1. Find, in simplest form,
   1. the value of \(k\)
   2. the value of \(p\) $$g ( x ) = \left( 2 + \frac { 16 } { x } \right) ( 1 + k x ) ^ { 16 }$$ Using the value of \(k\) found in part (a),
2. find the term in \(x ^ { 2 }\) in the expansion of \(\mathrm { g } ( x )\). $$\begin{aligned} u _ { 1 } & = 6
   u _ { n + 1 } & = k u _ { n } + 3 \end{aligned}$$ where \(k\) is a positive constant.
3. Find, in terms of \(k\), an expression for \(u _ { 3 }\) Given that \(\sum _ { n = 1 } ^ { 3 } u _ { n } = 117\)
4. find the value of \(k\).

2. (a) Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of $$( 1 + p x ) ^ { 9 }$$ where \(p\) is a constant. These first 3 terms are \(1,36 x\) and \(q x ^ { 2 }\), where \(q\) is a constant.
(b) Find the value of \(p\) and the value of \(q\).

1. The first three terms in ascending powers of \(x\) in the binomial expansion of \(( 1 + p x ) ^ { 12 }\) are given by

$$1 + 18 x + q x ^ { 2 }$$ where \(p\) and \(q\) are constants.
Find the value of \(p\) and the value of \(q\).

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

$$1 + A x + B x ^ { 2 } + B x ^ { 3 } + \ldots$$ where \(k\) is a positive constant and \(A\), \(B\) and \(n\) are positive integers.

1. By considering the coefficients of \(x ^ { 2 }\) and \(x ^ { 3 }\), show that \(3 = ( n - 2 ) k\). Given that \(A = 4\),
2. find the value of \(n\) and the value of \(k\).
   

   7. continued

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3 One of the terms in the binomial expansion of \(( 4 + a x ) ^ { 6 }\) is \(160 x ^ { 3 }\).

1. Find the value of \(a\).
2. Using this value of \(a\), find the first two terms in the expansion of \(( 4 + a x ) ^ { 6 }\) in ascending powers of \(x\).

5 The first four terms in the binomial expansion of \(( 3 + k x ) ^ { 5 }\), in ascending powers of \(x\), can be written as \(a + b x + c x ^ { 2 } + d x ^ { 3 }\).

1. State the value of \(a\).
2. Given that \(b = c\), find the value of \(k\).
3. Hence find the value of \(d\).

8. $$g ( x ) = ( 2 + a x ) ^ { 8 } \quad \text { where } a \text { is a constant }$$ Given that one of the terms in the binomial expansion of \(\mathrm { g } ( x )\) is \(3402 x ^ { 5 }\)

1. find the value of \(a\). Using this value of \(a\),
2. find the constant term in the expansion of $$\left( 1 + \frac { 1 } { x ^ { 4 } } \right) ( 2 + a x ) ^ { 8 }$$

11. The first 3 terms, in ascending powers of \(x\), in the binomial expansion of \(( 1 + k x ) ^ { 10 }\) are given by $$1 + 15 x + p x ^ { 2 }$$ where \(k\) and \(p\) are constants.
a. Find the value of \(k\)
b. Find the value of \(p\)
c. Given that, in the expansion of \(( 1 + k x ) ^ { 10 }\), the coefficient of \(x ^ { 4 }\) is \(q\), find the value of \(q\).

5

1. 1. Describe the geometrical transformation that maps the graph of \(y = \left( 1 + \frac { x } { 3 } \right) ^ { 6 }\) onto the graph of \(y = ( 1 + 2 x ) ^ { 6 }\).
   2. The curve \(y = \left( 1 + \frac { x } { 3 } \right) ^ { 6 }\) is translated by the vector \(\left[ \begin{array} { l } 3
      0 \end{array} \right]\) to give the curve \(y = \mathrm { g } ( x )\). Find an expression for \(\mathrm { g } ( x )\), simplifying your answer.
2. The first four terms in the binomial expansion of \(\left( 1 + \frac { x } { 3 } \right) ^ { 6 }\) are \(1 + a x + b x ^ { 2 } + c x ^ { 3 }\). Find the values of the constants \(a , b\) and \(c\), giving your answers in their simplest form.

2. (a) Write down the first four terms of the binomial expansion, in ascending powers of \(x\), of \(( 1 + a x ) ^ { n }\), where \(n > 2\). Given that, in this expansion, the coefficient of \(x\) is 8 and the coefficient of \(x ^ { 2 }\) is 30 ,
(b) find the value of \(n\) and the value of \(a\),
(c) find the coefficient of \(x ^ { 3 }\).

5. The expansion of \(( 2 - p x ) ^ { 6 }\) in ascending powers of \(x\), as far as the term in \(x ^ { 2 }\), is $$64 + A x + 135 x ^ { 2 }$$ Given that \(p > 0\), find the value of \(p\) and the value of \(A\).

5. (a) Write down the first 4 terms of the binomial expansion, in ascending powers of \(x\), of $$( 1 + a x ) ^ { n } , n > 2 .$$ Given that, in this expansion, the coefficient of \(x\) is 8 and the coefficient of \(x ^ { 2 }\) is 30 ,
(b) calculate the value of \(n\) and the value of \(a\),
(c) find the coefficient of \(x ^ { 3 }\).
[0pt] [P2 November 2003 Question 3]

4. The first three terms in the expansion in descending powers of \(x\) of $$\left( x + \frac { k } { x ^ { 2 } } \right) ^ { 15 } ,$$ where \(k\) is a constant, are $$x ^ { 15 } + 30 x ^ { 12 } + A x ^ { 9 } .$$

1. Find the values of \(k\) and \(A\).
2. Find the value of the term independent of \(x\) in the expansion.