Coefficient relationship between terms

3. (a) Find the first four terms, in ascending powers of \(x\), in the binomial expansion of \(( 1 + k x ) ^ { 6 }\), where \(k\) is a non-zero constant. Given that, in this expansion, the coefficients of \(x\) and \(x ^ { 2 }\) are equal, find
(b) the value of \(k\),
(c) the coefficient of \(x ^ { 3 }\).

3. (a) Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of \(( 1 + a x ) ^ { 10 }\), where \(a\) is a non-zero constant. Give each term in its simplest form. Given that, in this expansion, the coefficient of \(x ^ { 3 }\) is double the coefficient of \(x ^ { 2 }\),
(b) find the value of \(a\).

2. (a) Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of $$( 2 + k x ) ^ { 7 }$$ where \(k\) is a constant. Give each term in its simplest form. Given that the coefficient of \(x ^ { 2 }\) is 6 times the coefficient of \(x\),
(b) find the value of \(k\).

2. (a) Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of $$( 3 + b x ) ^ { 5 }$$ where \(b\) is a non-zero constant. Give each term in its simplest form. Given that, in this expansion, the coefficient of \(x ^ { 2 }\) is twice the coefficient of \(x\),
(b) find the value of \(b\).

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 \(d\).

5 The coefficient of \(x ^ { 3 }\) in the expansion of \(( 2 + a x ) ^ { 5 }\) is 10 times the coefficient of \(x ^ { 2 }\) in \(\left( 1 + \frac { a x } { 3 } \right) ^ { 4 }\). Find \(a\).

The coefficient of \(x^4\) in the expansion of \((x + a)^6\) is \(p\) and the coefficient of \(x^2\) in the expansion of \((ax + 3)^4\) is \(q\). It is given that \(p + q = 276\). Find the possible values of the constant \(a\). [4]

The coefficient of \(x^2\) in the expansion of \((1-4x)^6\) is 12 times the coefficient of \(x^2\) in the expansion of \((2+ax)^5\). Find the value of the positive constant \(a\). [3]

The coefficient of \(x^3\) in the expansion of \((3 + 2ax)^5\) is six times the coefficient of \(x^2\) in the expansion of \((2 + ax)^6\). Find the value of the constant \(a\). [4]

The coefficients of \(x\) and \(x^2\) in the expansion of \((2 + ax)^7\) are equal. Find the value of the non-zero constant \(a\). [3]

In the expansion of \((2 + ax)^6\), the coefficient of \(x^2\) is equal to the coefficient of \(x^3\). Find the value of the non-zero constant \(a\). [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]

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]

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