Find constants from coefficient conditions on terms

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. Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of \((1 + px)^9\), where \(p\) is a constant. [2]

The first 3 terms are 1, 36x and \(qx^2\), where \(q\) is a constant.

1. Find the value of \(p\) and the value of \(q\). [4]

The first four terms, in ascending powers of \(x\), of the binomial expansion of \((1 + kx)^n\) are $$1 + Ax + Bx^2 + Bx^3 + \ldots,$$ 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 \(3 = (n - 2) k\). [4]

Given that \(A = 4\),

1. find the value of \(n\) and the value of \(k\). [4]

$$f(x) = \left(1 + \frac{x}{k}\right)^n, \quad k, n \in \mathbb{N}, \quad n > 2.$$ Given that the coefficient of \(x^3\) is twice the coefficient of \(x^2\) in the binomial expansion of \(f(x)\),

1. prove that \(n = 6k + 2\). [3]

Given also that the coefficients of \(x^4\) and \(x^5\) are equal and non-zero,

1. form another equation in \(n\) and \(k\) and hence show that \(k = 2\) and \(n = 14\). [4]

Using these values of \(k\) and \(n\),

1. expand \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^5\). Give each coefficient as an exact fraction in its lowest terms [4]

The first three terms in the expansion, in ascending powers of \(x\), of \((1 + px)^n\), are \(1 - 18x + 36p^2x^2\). Given that \(n\) is a positive integer, find the value of \(n\) and the value of \(p\). [7]

$$f(x) = \left(1 + \frac{x}{k}\right)^n, \quad k, n \in \mathbb{N}, \quad n > 2.$$ Given that the coefficient of \(x^3\) is twice the coefficient of \(x^2\) in the binomial expansion of f(x),

1. prove that \(n = 6k + 2\). [3]

Given also that the coefficients of \(x^4\) and \(x^5\) are equal and non-zero,

1. form another equation in \(n\) and \(k\) and hence show that \(k = 2\) and \(n = 14\). [4]

Using these values of \(k\) and \(n\),

1. expand f(x) in ascending powers of \(x\), up to and including the term in \(x^5\). Give each coefficient as an exact fraction in its lowest terms [4]

The first three terms in the expansion, in ascending powers of \(x\), of \((1 + px)^n\), are \(1 - 18x + 36p^2x^2\). Given that \(n\) is a positive integer, find the value of \(n\) and the value of \(p\). [7]