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

The first 3 terms, in ascending powers of \(x\), in the binomial expansion of \((1 + kx)^{10}\) are given by $$1 + 15x + px^2$$ where \(k\) and \(p\) are constants.

1. Find the value of \(k\) [2]
2. Find the value of \(p\) [2]
3. Given that, in the expansion of \((1 + kx)^{10}\), the coefficient of \(x^4\) is \(q\), find the value of \(q\). [2]

Find the sum of all the integers from 1 to 999 inclusive that are not square or cube numbers. [5 marks]

In the binomial expansion of \((2 - 5x)^8\), find

1. the number of terms, [1]
2. the \(4^{\text{th}}\) term, when the expansion is in ascending powers of \(x\), [2]
3. the greatest positive coefficient. [3]

Find the term which is independent of \(x\) in the expansion of \(\frac{(2-3x)^5}{x^3}\). [4]

Use the binomial expansion to find, in ascending powers of \(x\), the first four terms in the expansion of $$\left(1 + \frac{3}{4}x\right)^6$$ simplifying each term. [4]

Find the coefficient of the \(x^4\) term in \((2 - 3x)^6\). [3]

In the question you must show detailed reasoning Given that the coefficients of \(x\), \(x^2\) and \(x^4\) in the expansion of \((1 + kx)^n\) are the consecutive terms of a geometric series, where \(n \geq 4\) and \(k\) is a positive constant

1. Show that $$k = \frac{6(n-1)}{(n-2)(n-3)}$$ [4]
2. For the case when \(k = \frac{7}{2}\), find the value of \(n\). [2]
3. Given that \(k = \frac{7}{5}\), \(n\) is a positive integer, and that the first term of the geometric series is the coefficient of \(x\), find the number of terms required for the sum of the geometric series to exceed \(1.12 \times 10^{12}\). [4]

Consider the binomial expansion of \(\left(1 + \frac{x}{5}\right)^n\) in ascending powers of \(x\), where \(n\) is a positive integer.

1. Write down the first four terms of the expansion, giving the coefficients as polynomials in \(n\). [1]

The coefficients of the second, third and fourth terms of the expansion are consecutive terms of an arithmetic sequence.

1. Show that \(n^3 - 33n^2 + 182n = 0\). [3]
2. Hence find the possible values of \(n\) and the corresponding values of the common difference. [3]

1. Find the first 4 terms, in ascending powers of \(x\), in the binomial expansion of $$(1 + kx)^{10}$$ where \(k\) is a non-zero constant. Write each coefficient as simply as possible. [3]

Given that in the expansion of \((1 + kx)^{10}\) the coefficient \(x^3\) is 3 times the coefficient of \(x\),

1. find the possible values of \(k\). [3]

Let \(C = \sum_{r=0}^{20} \binom{20}{r} \cos(r\theta)\). Show that \(C = 2^{20} \cos^{20}\left(\frac{1}{2}\theta\right) \cos(10\theta)\). [8]

Expand \((x - 2y)^5\). [3]

In all questions you must show all stages of your working, justifying solutions and not relying solely on calculator technology.

1. Find the first four terms, in ascending powers of \(x\), of the binomial expansion of \(\left(1+\frac{x}{2}\right)^7\), giving each coefficient in exact simplified form. [3]
2. Hence determine the coefficient of \(x\) in the expansion of $$\left(1+\frac{2}{x}\right)^2\left(1+\frac{x}{2}\right)^7.$$ [3]

In this question you must show detailed reasoning. Consider the expansion of \(\left(\frac{x^2}{2} + \frac{a}{x}\right)^6\). The constant term is 960. Find the possible values of \(a\). [4]