Find term or common difference

7. (a) An arithmetic series has a common difference of 7 . Given that the sum of the first 20 terms of the series is 530 , find

1. the first term of the series,
2. the smallest positive term of the series.
   (b) The terms of a sequence are given by $$u _ { n } = ( n + k ) ^ { 2 } , \quad n \geq 1 ,$$ where \(k\) is a positive constant.
   Given that \(u _ { 2 } = 2 u _ { 1 }\),
3. find the value of \(k\),
4. show that \(u _ { 3 } = 11 + 6 \sqrt { 2 }\).

8. (a) The first and third terms of an arithmetic series are 3 and 27 respectively.

1. Find the common difference of the series.
2. Find the sum of the first 11 terms of the series.
   (b) Find the sum of the integers between 50 and 150 which are divisible by 8 .

4 An arithmetic series has first term \(a\) and common difference \(d\).
The sum of the first 29 terms is 1102.

1. Show that \(a + 14 d = 38\).
2. The sum of the second term and the seventh term is 13 . Find the value of \(a\) and the value of \(d\).

5 An arithmetic progression has first term 5 and common difference 7.

1. Find the value of the 10th term.
2. Find the sum of the first 15 terms. The terms of the progression are given by \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\).
3. Evaluate \(\sum _ { n = 1 } ^ { 15 } \left( 2 x _ { n } + 1 \right)\).

An arithmetic progression has fourth term 15 and eighth term 25. Find the 30th term of the progression. [3]

An athlete prepares for a race by completing a practice run on each of 11 consecutive days. On each day after the first day he runs further than he ran on the previous day. The lengths of his 11 practice runs form an arithmetic sequence with first term \(a\) km and common difference \(d\) km. He runs 9 km on the 11th day, and he runs a total of 77 km over the 11 day period. Find the value of \(a\) and the value of \(d\). [7]

The 11th term of an arithmetic progression is 1. The sum of the first 10 terms is 120. Find the 4th term. [5]

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]

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]

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]

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 \(u_1, u_2, u_3 \ldots\) is defined by \(u_1 = 7\) and \(u_{n+1} = u_n + 4\) for \(n \geq 1\).

1. State what type of sequence this is. [1]
2. Find \(u_{17}\). [2]

An arithmetic series has first term \(a\) and common difference \(d\). Given that the sum of the first 9 terms is 54

1. show that $$a + 4d = 6$$ [2]

Given also that the 8th term is half the 7th term,

1. find the values of \(a\) and \(d\). [4]

The 11th term of an arithmetic progression is 1. The sum of the first 10 terms is 120. Find the 4th term. [5]

An arithmetic series has first term \(a\) and common difference \(d\). The sum of the first 29 terms is 1102.

1. Show that \(a + 14d = 38\). (3 marks)
2. The sum of the second term and the seventh term is 13. Find the value of \(a\) and the value of \(d\). (4 marks)