The first term of a progression is \(4x\) and the second term is \(x^2\).
For the case where the progression is arithmetic with a common difference of 12, find the possible values of \(x\) and the corresponding values of the third term.
The first term of an arithmetic progression is \(-2222\) and the common difference is 17. Find the value of the first positive term.
The first, second and last terms in an arithmetic progression are 56, 53 and -22 respectively. Find the sum of all the terms in the progression.
The first, second and third terms of an arithmetic progression are \(a, 2a\) and \(a^2\) respectively, where \(a\) is a positive constant.
Find the sum of the first 50 terms of the progression.
A circle is divided into 5 sectors in such a way that the angles of the sectors are in arithmetic progression. Given that the angle of the largest sector is 4 times the angle of the smallest sector, find the angle of the largest sector.
The sum, \(S_n\), of the first \(n\) terms of an arithmetic progression is given by \(S_n = 32n - n^2\). Find the first term and the common difference.
An arithmetic progression has first term 7. The nth term is 84 and the (3n)th term is 245. Find the value of n.
An arithmetic progression has first term \(a\) and common difference \(d\). It is given that the sum of the first 200 terms is 4 times the sum of the first 100 terms.
(i) Find \(d\) in terms of \(a\).
(ii) Find the 100th term in terms of \(a\).
In an arithmetic progression, the fifth term is 197 and the sum of the first ten terms is 2040. Find the common difference.
An athlete runs the first mile of a marathon in 5 minutes. His speed reduces in such a way that each mile takes 12 seconds longer than the preceding mile.
(i) Given that the nth mile takes 9 minutes, find the value of n.
(ii) Assuming that the length of the marathon is 26 miles, find the total time, in hours and minutes, to complete the marathon.
In an arithmetic progression the sum of the first ten terms is 400 and the sum of the next ten terms is 1000. Find the common difference and the first term.
In an arithmetic progression, the sum, \(S_n\), of the first \(n\) terms is given by \(S_n = 2n^2 + 8n\). Find the first term and the common difference of the progression.
The first and last terms of an arithmetic progression are 12 and 48 respectively. The sum of the first four terms is 57. Find the number of terms in the progression.
A circle is divided into n sectors in such a way that the angles of the sectors are in arithmetic progression. The smallest two angles are 3° and 5°. Find the value of n.
An arithmetic progression has first term 4 and common difference \(d\). The sum of the first \(n\) terms of the progression is 5863.
(a) Show that \((n-1)d = \frac{11726}{n} - 8\).
(b) Given that the \(n\)th term is 139, find the values of \(n\) and \(d\), giving the value of \(d\) as a fraction.
The first term of an arithmetic progression is 61 and the second term is 57. The sum of the first n terms is n. Find the value of the positive integer n.
In an arithmetic progression, the sum of the first n terms, denoted by Sn, is given by
\(S_n = n^2 + 8n\).
Find the first term and the common difference.
An arithmetic progression contains 25 terms and the first term is -15. The sum of all the terms in the progression is 525. Calculate
(i) the common difference of the progression,
(ii) the last term in the progression,
(iii) the sum of all the positive terms in the progression.
The sixth term of an arithmetic progression is 23 and the sum of the first ten terms is 200. Find the seventh term.
An arithmetic progression is such that the eighth term is three times the third term. Show that the sum of the first eight terms is four times the sum of the first four terms.