1.04h Arithmetic sequences: nth term and sum formulae

An arithmetic progression has tenth term 11.1 and fiftieth term 7.1. Find the first term and the common difference. Find also the sum of the first fifty terms of the progression. [5]

Find the second and third terms in the sequence given by $$u_1 = 5,$$ $$u_{n+1} = u_n + 3.$$ Find also the sum of the first 50 terms of this sequence. [4]

In an arithmetic progression, the second term is 11 and the sum of the first 40 terms is 3030. Find the first term and the common difference. [5]

The third term of an arithmetic progression is 24. The tenth term is 3. Find the first term and the common difference. Find also the sum of the 21st to 50th terms inclusive. [5] Simplify

An arithmetic progression has first term 7 and third term 12.

1. Find the 20th term of this progression. [2]
2. Find the sum of the 21st to the 50th terms inclusive of this progression. [3]

Triangle \(ABC\), with \(BC = a\), \(AC = b\) and \(AB = c\) is inscribed in a circle. Given that \(AB\) is a diameter of the circle and that \(a^2\), \(b^2\) and \(c^2\) are three consecutive terms of an arithmetic progression (arithmetic series),

1. express \(b\) and \(c\) in terms of \(a\), [4]
2. verify that \(\cot A\), \(\cot B\) and \(\cot C\) are consecutive terms of an arithmetic progression. [3]

In an acute-angled triangle \(PQR\) the sides \(QR\), \(PR\) and \(PQ\) have lengths \(p\), \(q\) and \(r\) respectively.

1. Prove that $$\frac{p}{\sin P} = \frac{q}{\sin Q} = \frac{r}{\sin R}.$$ [3]

Given now that triangle \(PQR\) is such that \(p^2\), \(q^2\) and \(r^2\) are three consecutive terms of an arithmetic progression,

1. use the cosine rule to prove that $$\frac{2\cos Q}{q} = \frac{\cos P}{p} + \frac{\cos R}{r}.$$ [6]
2. Using the results given in parts \((c)\) and \((d)\), prove that \(\cot P\), \(\cot Q\) and \(\cot R\) are consecutive terms in an arithmetic progression. [3]

The first and second terms of an arithmetic series are 200 and 197.5 respectively. The sum to \(n\) terms of the series is \(S_n\). Find the largest positive value of \(S_n\). [5]

An arithmetic progression has first term \(2\) and common difference \(d\), where \(d \neq 0\). The first, third and thirteenth terms of this progression are also the first, second and third terms, respectively, of a geometric progression. By determining \(d\), show that the arithmetic progression is an increasing sequence. [5]

The positive integers \(x\), \(y\) and \(z\) are the first, second and third terms, respectively, of an arithmetic progression with common difference \(-4\). Also, \(x\), \(\frac{15}{y}\) and \(z\) are the first, second and third terms, respectively, of a geometric progression.

1. Show that \(y\) satisfies the equation \(y^4 - 16y^2 - 225 = 0\). [4]
2. Hence determine the sum to infinity of the geometric progression. [4]

The first, third and fourth terms of an arithmetic progression are \(u_1\), \(u_3\) and \(u_4\) respectively, where $$u_1 = 2 \sin \theta, \quad u_3 = -\sqrt{3} \cos \theta, \quad u_4 = \frac{7}{3} \sin \theta,$$ and \(\frac{1}{2}\pi < \theta < \pi\).

1. Determine the exact value of \(\theta\). [3]
2. Hence determine the value of \(\sum_{r=1}^{100} u_r\). [3]

An arithmetic sequence has first term \(a\) and common difference \(d\). The sum of the first 16 terms of the sequence is 260

1. Show that \(4a + 30d = 65\) [2 marks]
2. Given that the sum of the first 60 terms is 315, find the sum of the first 41 terms. [3 marks]
3. \(S_n\) is the sum of the first \(n\) terms of the sequence. Explain why the value you found in part (b) is the maximum value of \(S_n\) [2 marks]

The positive integer \(k\) is such that $$\sum_{r=1}^{k} (3r - k) = 90$$ Find the value of \(k\). [3 marks]

Find the sum of all the integers from 1 to 999 inclusive that are not square or cube numbers. [5 marks]

Find seven numbers which are in arithmetic progression such that the last term is 71 and the sum of all of the numbers is 329. [5]

The 12th term of an arithmetic series is 41 and the sum of the first 16 terms is 488. Find the first term and the common difference of the series. [5]

Showing all your working, evaluate

1. \(\sum_{r=3}^{50} (4r + 5)\) [4]
2. \(\sum_{r=2}^{\infty} \left(540 \times \left(\frac{1}{3}\right)^r\right)\). [3]

The lengths of the sides of a fifteen-sided plane figure form an arithmetic sequence. The perimeter of the figure is 270 cm and the length of the largest side is eight times that of the smallest side. Find the length of the smallest side. [4]

A sequence \(u_1, u_2, u_3, ...\) is defined by \(u_n = 3n - 1\), for \(n \geq 1\).

1. Find the values of \(u_1, u_2, u_3\). [1]
2. Find $$\sum_{n=1}^{40} u_n$$ [3]