(a) Find the first three terms in the expansion of
\(\left(x^2-\frac4{x^2}\right)^{10}\)
in descending powers of \(x\). Give each term in its simplest form.
(b) Hence find the coefficient of \(x^{16}\) in the expansion of
\(\left(x^2-\frac4{x^2}\right)^{10}\left(x^2+\frac2{x^2}\right)^2.\)
In the expansion of
\(\left(ax+\frac{b}{x^2}\right)^9,\)
where \(a\) and \(b\) are constants with \(a\gt 0\), the term independent of \(x\) is \(-145\,152\) and the coefficient of \(x^6\) is \(-6912\). Show that \(a^2b=-12\) and find the value of \(a\) and the value of \(b\).
(a)(i) Find the first three terms in the expansion of
\(\left(1+\frac{x}{7}\right)^5\)
in ascending powers of \(x\). Simplify the coefficient of each term.
(a)(ii) The expansion of
\(7(1+x)^n\left(1+\frac{x}{7}\right)^5,\)
where \(n\) is a positive integer, is written in ascending powers of \(x\). The first two terms in the expansion are \(7+89x\). Find the value of \(n\).
(b) In the expansion of \((k-2x)^8\), where \(k\) is a constant, the coefficient of \(x^4\) divided by the coefficient of \(x^2\) is \(\frac58\). The coefficient of \(x\) is positive. Form an equation and hence find the value of \(k\).
(a) It is given that the first four terms, in ascending powers of \(x\), in the expansion of \(\left(1-\frac{x}{2}\right)^n\) can be written in the form
\(1-8x+px^2+qx^3\),
where \(n\), \(p\) and \(q\) are integers. Find the values of \(n\), \(p\) and \(q\).
(b) Find the term independent of \(x\) in the expansion of \(\left(\frac{2}{x^2}+\frac{x}{3}\right)^6\), giving your answer as a rational number.
The first three terms, in descending powers of \(x\), in the expansion of
\(\displaystyle \left(2x^2-\frac1{4x}\right)^n\)
can be written in the form
\(256x^{16}+ax^{13}+bx^c\),
where \(n\), \(a\), \(b\) and \(c\) are integers. Find the values of \(n\), \(a\), \(b\) and \(c\).
(a)(i) Use the binomial theorem to expand \((1+3x)^7\) in ascending powers of \(x\), as far as the term in \(x^3\). Simplify each term.
(a)(ii) Show that your expansion from part (a)(i) gives the value of \(1.03^7\) as \(1.23\) to \(2\) decimal places.
(b) Find the term independent of \(x\) in the expansion of
\(\left(\frac{x^4}{2}+\frac{2}{x}\right)^{15}.\)
The first three terms, in ascending powers of \(x\), in the expansion of
\(\left(1+\frac{x}{6}\right)^{12}(2-3x)^3\)
can be written in the form \(8+px+qx^2\), where \(p\) and \(q\) are constants. Find the values of \(p\) and \(q\).
(a) Find the rational numbers \(a\), \(b\) and \(c\), such that the first three terms, in descending powers of \(x\), in the expansion of
\(\left(3x^2-\frac{1}{9x}\right)^5\)
can be written in the form \(ax^{10}+bx^7+cx^4\).
(b) Hence find the coefficient of \(x^4\) in the expansion of
\(\left(3x^2-\frac{1}{9x}\right)^5\left(1+\frac{1}{x^3}\right)^2.\)
(a) (i) Write down the first three terms, in ascending powers of \(x\), in the expansion of \((1+4x)^n\), where \(n\) is a positive integer.
(ii) In the expansion of \((1+4x)^n(1-4x)\), the coefficient of \(x^2\) is 6032. Find the value of \(n\).
(b) Find the term independent of \(x\) in the expansion of \(\left(\frac{x}{2}-\frac{8}{x^4}\right)^{10}\).
The first three terms, in ascending powers of \(x\), in the expansion of
\(\left(1-\frac{2x}{9}\right)^{18}(1+3x)^3\)
are written in the form
\(1+ax+bx^2,\)
where \(a\) and \(b\) are constants. Find the exact values of \(a\) and \(b\).
The first three terms, in descending powers of \(x\), of the expansion of
\(\left(ax+\frac25\right)^5\left(1-\frac{b}{x}\right)^2\)
can be written as
\(32x^5-160x^4+cx^3,\)
where \(a\), \(b\) and \(c\) are constants. Find the exact values of \(a\), \(b\) and \(c\).
The first four terms in the expansion of \((3+ax)^4\), in ascending powers of \(x\), are
\(81+bx+cx^2+\frac32x^3.\)
Find the values of \(a\), \(b\) and \(c\).
(a) In the expansion of
\(\left(2k-\frac{x}{k}\right)^5,\)
where \(k\) is a constant, the coefficient of \(x^2\) is \(160\). Find the value of \(k\).
(b)
(i) Find, in ascending powers of \(x\), the first 3 terms in the expansion of \((1+3x)^6\), simplifying the coefficient of each term.
(ii) When \((1+3x)^6(a+x)^2\) is written in ascending powers of \(x\), the first three terms are \(4+68x+bx^2\), where \(a\) and \(b\) are constants. Find the value of \(a\) and of \(b\).
The first 3 terms in the expansion of
\((a+x)^3\left(1-\frac{x}{3}\right)^5\)
in ascending powers of \(x\), can be written in the form
\(27+bx+cx^2,\)
where \(a\), \(b\) and \(c\) are integers. Find the values of \(a\), \(b\) and \(c\).
(a) Find the first three non-zero terms in the expansion of
\(\left(2-\frac{x^2}{4}\right)^6\)
in ascending powers of \(x\). Simplify each term.
(b) Hence find the term independent of \(x\) in the expansion of
\(\left(2-\frac{x^2}{4}\right)^6\left(3-\frac1{x^2}\right)^2.\)
Using the binomial theorem, expand
\((1+e^{2x})^4,\)
simplifying each term.
The first three terms, in ascending powers of \(x\), in the expansion of \((2+ax)^n\) can be written as \(64+bx+cx^2\), where \(n\), \(a\), \(b\) and \(c\) are constants.
(a) Find the value of \(n\).
(b) Show that \(5b^2=768c\).
(c) Given that \(b=12\), find the exact value of \(a\) and of \(c\).
(a) Find the first three terms, in ascending powers of \(x^2\), in the expansion of
\(\left(\frac12-\frac23x^2\right)^8.\)
Write your coefficients as rational numbers.
(b) Find the coefficient of \(x^2\) in the expansion of
\(\left(\frac12-\frac23x^2\right)^8\left(2x+\frac1x\right)^2.\)
(a) Expand \((2-3x)^4\), evaluating all of the coefficients.
(b) The sum of the first three terms in ascending powers of \(x\) in the expansion of
\((2-3x)^4\left(1+\frac ax\right)\)
is
\(\frac{32}{x}+b+cx,\)
where \(a\), \(b\) and \(c\) are integers. Find the values of \(a\), \(b\) and \(c\).
The first 3 terms in the expansion of \((3-ax)^5\), in ascending powers of \(x\), can be written in the form
\(b-81x+cx^2.\)
Find the value of each of \(a\), \(b\) and \(c\).