The first three terms in the expansion of
\(\left(3-\frac{x}{6}\right)^n\)
are \(81+ax+bx^2\). Find \(n\), \(a\), and \(b\).
(i) Given that \(a\) is a constant, expand \((2+ax)^4\), in ascending powers of \(x\), simplifying each term of your expansion.
Given also that the coefficient of \(x^2\) is equal to the coefficient of \(x^3\),
(ii) show that \(a=3\),
(iii) use your expansion to show that the value of \(1.97^4\) is \(15.1\) to 1 decimal place.
The first three terms of the binomial expansion of \((2-ax)^n\) are
\(64-16bx+100bx^2.\)
Find the value of each of the integers \(n\), \(a\) and \(b\).
(i) Find, in ascending powers of \(x\), the first 3 terms in the expansion of \(\left(2-\dfrac{x^2}{4}\right)^5\).
(ii) Hence find the term independent of \(x\) in the expansion of \(\left(2-\dfrac{x^2}{4}\right)^5\left(\dfrac1x-\dfrac3{x^2}\right)^2\).
(i) Find, in ascending powers of \(x\), the first 3 terms in the expansion of \(\left(2-\dfrac{x^2}{4}\right)^6\).
(ii) Hence find the coefficient of \(x^2\) in the expansion of \(\left(2-\dfrac{x^2}{4}\right)^6\left(\dfrac1x+x\right)^2\).
(i) Expand \((1+x)^4\), simplifying all coefficients.
(ii) Expand \((6-x)^4\), simplifying all coefficients.
(iii) Hence express \((6-x)^4-(1+x)^4=175\) in the form \(ax^3+bx^2+cx+d=0\), where \(a\), \(b\), \(c\), and \(d\) are integers.
(iv) Show that \(x=2\) is a solution of the equation in part (iii) and show that this equation has no other real roots.
An arithmetic progression has common difference \(d\). The 3rd term of this progression is \(10\).
(a) Write down expressions for the 1st term and the 2nd term of this progression. Give your answers in terms of \(d\) only.
(b) When each of the first 3 terms is squared, the sum of these squares is \(140\). There are two possible values for \(d\).
Using your answer to part (a), find the sum of the first \(200\) terms of the progression with the smaller value of \(d\).
Two arithmetic progressions, \(A\) and \(B\), each have 100 terms. Their terms are denoted by \(a_{1}, a_{2}, a_{3}, a_{4}, \ldots a_{100}\) and \(b_{1}, b_{2}, b_{3}, b_{4}, \ldots b_{100}\) respectively.
It is given that \(a_{1}=b_{100}=1\) and \(a_{100}=b_{1}=298\). (a) Find \(n\) such that \(a_{n}-b_{n}=45\). (b) Find the smallest \(m\) such that \(a_{m}\gt 2 b_{m}\).
An arithmetic progression is such that the fourth term is \(25\) and the ninth term is \(50\).
(a) Find the first term and the common difference.
(b) Find the least number of terms for which the sum of the progression is greater than \(25000\).
The first three terms of an arithmetic progression are \(\lg x\), \(\lg x^5\), \(\lg x^9\), where \(x\gt 0\).
(a) Show that the sum to \(n\) terms of this arithmetic progression can be written as
\(n(pn-1)\lg x,\)
where \(p\) is an integer.
(b) Hence find the value of \(n\) for which the sum to \(n\) terms is equal to \(4950\lg x\).
(c) Given that this sum to \(n\) terms is also equal to \(-14850\), find the exact value of \(x\).
An arithmetic progression has third term \(10\), and the sum of the first \(8\) terms is \(116\).
(a) Find the first term and the common difference.
(b) Find the sum of \(19\) terms of the progression, starting with the twelfth term.
An arithmetic progression has first term \(a\) and common difference \(d\). The third term is \(13\) and the tenth term is \(41\).
(a) Find the value of \(a\) and of \(d\).
(b) Find the number of terms required to give a sum of \(2555\).
(c) Given that \(S_n\) is the sum to \(n\) terms, show that
\(S_{2k}-S_k=3k(1+2k).\)
The seventh term of an arithmetic progression is \(158\), and the tenth term is \(149\).
(a) Find the first term and the common difference of the progression.
(b) Find the least value of \(n\) for which the sum of the first \(n\) terms is negative.
The first term of a geometric progression is \(10\). This geometric progression has a positive common ratio \(r\).
The first term of an arithmetic progression is also \(10\). This arithmetic progression has a negative common difference \(d\).
The second term of the geometric progression is the same as the fourth term of the arithmetic progression.
The third term of the geometric progression is the same as the sixth term of the arithmetic progression.
(a) Find the values of \(r\) and \(d\).
(b) Determine whether the geometric progression has a sum to infinity.
(a) Suzma is training for a marathon. In the first week she runs 10 km . Then each week she runs a distance that is \(10 \%\) greater than the week before.
The total distance that Suzma has run by the end of \(n\) whole weeks is more than 200 km . Find the smallest possible value of \(n\). (b) A geometric progression has 1st term \(a\) and common ratio \(r\), where \(a \neq 0\) and \(r \neq 1\). The 1st, 2nd and 3rd terms of the geometric progression are the 1st, 3rd and 7th terms of an arithmetic progression. Find the value of \(r\).
(a) A geometric progression has third term \(4.5\) and sixth term \(15.1875\). Find the first term and the common ratio.
(b) Find the sum of ten terms of the progression, starting with the sixteenth term. Give your answer to the nearest integer.
A geometric progression is such that its sum to \(4\) terms is \(17\) times its sum to \(2\) terms. It is given that the common ratio of this geometric progression is positive and not equal to \(1\).
(a) Find the common ratio of this geometric progression.
(b) Given that the \(6\)th term of the geometric progression is \(64\), find the first term.
(c) Explain why this geometric progression does not have a sum to infinity.
A geometric progression has a first term of \(3\) and a second term of \(2.4\). For this progression, find
(a) the sum of the first \(8\) terms,
(b) the sum to infinity,
(c) the least number of terms for which the sum is greater than \(95\%\) of the sum to infinity.
The sum of the first two terms of a geometric progression is \(9\). The sum to infinity of this geometric progression is \(25\).
(a) Find the possible values of the common ratio of this geometric progression.
(b) Find the first term of this geometric progression for each possible value of the common ratio.
A geometric progression has a 4th term of \(\frac{8k^6}{27}\) and a 6th term of \(\frac{32k^{10}}{243}\), where \(k\) is a constant. The common ratio of this geometric progression is positive.
(a) Find the common ratio in terms of \(k\) and the value of the first term of this geometric progression.
(b) Given that this geometric progression has a sum to infinity of \(3\), find the possible values of \(k\).