The coefficient of \(\frac{1}{x}\) in the expansion of \(\left( kx + \frac{1}{x} \right)^5 + \left( 1 - \frac{2}{x} \right)^8\) is 74.
Find the value of the positive constant \(k\).
The coefficient of \(\frac{1}{x}\) in the expansion of \(\left( 2x + \frac{a}{x^2} \right)^5\) is 720.
(a) Find the possible values of the constant \(a\).
(b) Hence find the coefficient of \(\frac{1}{x^7}\) in the expansion.
The coefficient of \(x^2\) in the expansion of \((4 + ax)\left(1 + \frac{x}{2}\right)^6\) is 3. Find the value of the constant \(a\).
The term independent of x in the expansion of \(\left( 2x + \frac{k}{x} \right)^6\), where k is a constant, is 540.
(i) Find the value of k.
(ii) For this value of k, find the coefficient of x2 in the expansion.
The coefficient of \(x^3\) in the expansion of \((1 - px)^5\) is \(-2160\). Find the value of the constant \(p\).
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 \(a\).
The coefficient of \(x^2\) in the expansion of \(\left( 2 + \frac{x}{2} \right)^6 + (a + x)^5\) is 330. Find the value of the constant \(a\).
The coefficients of x and x2 in the expansion of (2 + ax)7 are equal. Find the value of the non-zero constant a.
The coefficients of x2 and x3 in the expansion of (3 - 2x)6 are a and b respectively. Find the value of \(\frac{a}{b}\).
In the expansion of \(\left( \frac{1}{ax} + 2ax^2 \right)^5\), the coefficient of \(x\) is 5. Find the value of the constant \(a\).
The coefficient of \(x^3\) in the expansion of \((1 - 3x)^6 + (1 + ax)^5\) is 100. Find the value of the constant \(a\).
In the expansion of \((3 - 2x)\left(1 + \frac{x}{2}\right)^n\), the coefficient of \(x\) is 7. Find the value of the constant \(n\) and hence find the coefficient of \(x^2\).
In the expansion of \((x + 2k)^7\), where \(k\) is a non-zero constant, the coefficients of \(x^4\) and \(x^5\) are equal. Find the value of \(k\).
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 \(a\).
In the expansion of \((2 + ax)^7\), the coefficient of \(x\) is equal to the coefficient of \(x^2\). Find the value of the non-zero constant \(a\).
In the expansion of \(\left(x^2 - \frac{a}{x}\right)^7\), the coefficient of \(x^5\) is \(-280\). Find the value of the constant \(a\).
(a) Give the complete expansion of \(\left( x + \frac{2}{x} \right)^5\).
(b) In the expansion of \((a + bx^2) \left( x + \frac{2}{x} \right)^5\), the coefficient of \(x\) is zero and the coefficient of \(\frac{1}{x}\) is 80. Find the values of the constants \(a\) and \(b\).
The coefficient of \(x^3\) in the expansion of \((a + x)^5 + (2 - x)^6\) is 90. Find the value of the positive constant \(a\).
The coefficient of \(x^2\) in the expansion of \(\left( k + \frac{1}{3}x \right)^5\) is 30. Find the value of the constant \(k\).
The coefficient of \(x^3\) in the expansion of \((a+x)^5 + (1-2x)^6\), where \(a\) is positive, is 90. Find the value of \(a\).