(i) Find the first 3 terms in the expansion of \((2-x)^6\) in ascending powers of \(x\).
(ii) Given that the coefficient of \(x^2\) in the expansion of \((1 + 2x + ax^2)(2-x)^6\) is 48, find the value of the constant \(a\).
(i) Find the first 3 terms in the expansion of \((2 + 3x)^5\) in ascending powers of \(x\).
(ii) Hence find the value of the constant \(a\) for which there is no term in \(x^2\) in the expansion of \((1 + ax)(2 + 3x)^5\).
(i) Find the first 3 terms in the expansion, in ascending powers of \(x\), of \((2 + x^2)^5\).
(ii) Hence find the coefficient of \(x^4\) in the expansion of \((1 + x^2)^2(2 + x^2)^5\).
The first three terms in the expansion of \((2+ax)^n\), in ascending powers of \(x\), are 32 - 40x + bx^2. Find the values of the constants \(n, a\) and \(b\).
(i) Find the first 3 terms in the expansion of \((2-x)^6\) in ascending powers of \(x\).
(ii) Find the value of \(k\) for which there is no term in \(x^2\) in the expansion of \((1+kx)(2-x)^6\).
(a) Find the first three terms in ascending powers of x of the expansion of \((1 + 2x)^5\).
(b) Find the first three terms in ascending powers of x of the expansion of \((1 - 3x)^4\).
(c) Hence find the coefficient of \(x^2\) in the expansion of \((1 + 2x)^5(1 - 3x)^4\).
(a) Expand \(\left( 1 - \frac{1}{2x} \right)^2\).
(b) Find the first four terms in the expansion, in ascending powers of \(x\), of \((1 + 2x)^6\).
(c) Hence find the coefficient of \(x\) in the expansion of \(\left( 1 - \frac{1}{2x} \right)^2 (1 + 2x)^6\).
(a) Write down the first four terms of the expansion, in ascending powers of \(x\), of \((a-x)^6\).
(b) Given that the coefficient of \(x^2\) in the expansion of \(\left(1 + \frac{2}{ax}\right)(a-x)^6\) is \(-20\), find in exact form the possible values of the constant \(a\).
(a) Find the first three terms in the expansion of \((3 - 2x)^5\) in ascending powers of \(x\).
(b) Hence find the coefficient of \(x^2\) in the expansion of \((4 + x)^2(3 - 2x)^5\).
(a) Find the first three terms in the expansion, in ascending powers of \(x\), of \((1 + x)^5\).
(b) Find the first three terms in the expansion, in ascending powers of \(x\), of \((1 - 2x)^6\).
(c) Hence find the coefficient of \(x^2\) in the expansion of \((1 + x)^5 (1 - 2x)^6\).
(i) Expand \((1+y)^6\) in ascending powers of \(y\) as far as the term in \(y^2\).
(ii) In the expansion of \((1 + (px - 2x^2))^6\) the coefficient of \(x^2\) is 48. Find the value of the positive constant \(p\).
(i) In the binomial expansion of \(\left( 2x - \frac{1}{2x} \right)^5\), the first three terms are \(32x^5 - 40x^3 + 20x\). Find the remaining three terms of the expansion.
(ii) Hence find the coefficient of \(x\) in the expansion of \((1 + 4x^2) \left( 2x - \frac{1}{2x} \right)^5\).
(a) Find the coefficient of \(x^2\) in the expansion of \(\left(x - \frac{2}{x}\right)^6\).
(b) Find the coefficient of \(x^2\) in the expansion of \((2 + 3x^2)\left(x - \frac{2}{x}\right)^6\).
(i) Find the coefficients of \(x^2\) and \(x^3\) in the expansion of \((2-x)^6\).
(ii) Find the coefficient of \(x^3\) in the expansion of \((3x+1)(2-x)^6\).
Find the coefficient of x in the expansion of \(\left(x^2 - \frac{2}{x}\right)^5\).
Find the coefficient of \(x^2\) in the expansion of \((1 + x^2) \left( \frac{x}{2} - \frac{4}{x} \right)^6\).
(i) Find the coefficient of \(x^8\) in the expansion of \((x + 3x^2)^4\).
(ii) Find the coefficient of \(x^8\) in the expansion of \((x + 3x^2)^5\).
(iii) Hence find the coefficient of \(x^8\) in the expansion of \([1 + (x + 3x^2)]^5\).
Find the coefficient of \(x^2\) in the expansion of
(i) \(\left( 2x - \frac{1}{2x} \right)^6\),
(ii) \((1 + x^2) \left( 2x - \frac{1}{2x} \right)^6\).
Find the coefficient of \(x^3\) in the expansion of \(\left( 2 - \frac{1}{2}x \right)^7\).
Find the coefficient of \(x^6\) in the expansion of \(\left( 2x^3 - \frac{1}{x^2} \right)^7\).