1.04a Binomial expansion: (a+b)^n for positive integer n

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]

1. Find the first four terms, in ascending powers of \(x\), in the bionomial expansion of \((1 + kx)^8\), where \(k\) is a non-zero constant. [3]

Given that, in this expansion, the coefficients of \(x\) and \(x^2\) are equal, find

1. the value of \(k\), [2]
2. the coefficient of \(x^3\). [1]

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]

1. Write down the first four terms of the binomial expansion, in ascending powers of \(x\), of \((1 + 3x)^n\), where \(n > 2\). [2]

Given that the coefficient of \(x^3\) in this expansion is ten times the coefficient of \(x^2\),

1. find the value of \(n\), [2]
2. find the coefficient of \(x^4\) in the expansion. [2]

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]

Find the coefficient of \(x^3\) in the expansion of \((2 + 3x)^5\). [4]

Find the binomial expansion of \((2 + x)^4\), writing each term as simply as possible. [4]

Calculate \(^6C_3\). Find the coefficient of \(x^3\) in the expansion of \((1 - 2x)^6\). [4]

Expand \((1 + \frac{1}{2}x)^4\), simplifying the coefficients. [4]

Find the coefficient of \(x^4\) in the binomial expansion of \((5 + 2x)^6\). [4]

Find the coefficient of \(x^3\) in the binomial expansion of \((2 - 4x)^5\). [4]

The expansion of \((2 - px)^6\) in ascending powers of \(x\), as far as the term in \(x^2\), is $$64 + Ax + 135x^2.$$ Given that \(p > 0\), find the value of \(p\) and the value of \(A\). [7]

The first three terms in the expansion, in ascending powers of x, of (1 + px)ⁿ, are 1 - 18x + 36p²x². Given that n is a positive integer, find the value of n and the value of p. [7]

1. Write down the first four terms of the binomial expansion, in ascending powers of x, of \((1 + ax)^n\), where \(n > 2\). [2]

Given that, in this expansion, the coefficient of x is 8 and the coefficient of x² is 30,

1. find the value of n and the value of a, [4]
2. find the coefficient of x³. [2]

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

1. prove that \(n = 6k + 2\). Given also that the coefficients of \(x^4\) and \(x^5\) are equal and non-zero, [3]
2. form another equation in \(n\) and \(k\) and hence show that \(k = 2\) and \(n = 14\). Using these values of \(k\) and \(n\), [4]
3. 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]

$$f(x) = \left(1 + \frac{x}{k}\right)^n, \quad k, n \in \mathbb{N}, \quad n > 2.$$ 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]

The expansion of \((2 - px)^6\) in ascending powers of \(x\), as far as the term in \(x^2\), is $$64 + Ax + 135x^2.$$ Given that \(p > 0\), find the value of \(p\) and the value of \(A\). [7]