Find term or common difference

2 The sum of the first 20 terms of an arithmetic progression is 405 and the sum of the first 40 terms is 1410 . Find the 60th term of the progression.

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

3 Ara rith etic p og essich s first term 7 Th \(n\)th erm is \& d (3n)ttl erm is \% Fid b lue \(6 n\).

3 The ninth term of an arithmetic progression is 22 and the sum of the first 4 terms is 49 .

1. Find the first term of the progression and the common difference. The \(n\)th term of the progression is 46 .
2. Find the value of \(n\).

3 The 12th term of an arithmetic progression is 17 and the sum of the first 31 terms is 1023. Find the 31st term.

5 In an arithmetic progression the first term is \(a\) and the common difference is 3 . The \(n\)th term is 94 and the sum of the first \(n\) terms is 1420 . Find \(n\) and \(a\).

4. The \(4 ^ { \text {th } }\) term of an arithmetic sequence is 3 and the sum of the first 6 terms is 27 Find the first term and the common difference of this sequence.

7. 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 \(a \mathrm {~km}\) and common difference \(d \mathrm {~km}\). 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 \(d\).

1 The 20th term of an arithmetic progression is 10 and the 50th term is 70 .

1. Find the first term and the common difference.
2. Show that the sum of the first 29 terms is zero.

7

1. In an arithmetic progression, the first term is 12 and the sum of the first 70 terms is 12915 . Find the common difference.
2. In a geometric progression, the second term is - 4 and the sum to infinity is 9 . Find the common ratio.

8 The 7th term of an arithmetic progression is 6. The sum of the first 10 terms of the progression is 30. Find the 5th term of the progression.

1. (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 \(u _ { 3 } = 11 + 6 \sqrt { 2 }\).

1 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.

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

3 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 21 st to 50 th terms inclusive. Simplify

1. Find the 51st term of the sequence given by $$\begin{aligned} u _ { 1 } & = 5 \\ u _ { n + 1 } & = u _ { n } + 4 \end{aligned}$$
2. Find the sum to infinity of the geometric progression which begins $$5 \quad 2 \quad 0.8 \quad \ldots .$$

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

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

1 The 7th term of an arithmetic progression is 6. The sum of the first 10 terms of the progression is 30. Find the 5th term of the progression.

2 The tenth term of an arithmetic progression is equal to twice the fourth term. The twentieth term of the progression is 44 .

1. Find the first term and the common difference.
2. Find the sum of the first 50 terms.

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

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

6 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.

10 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.

3 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.

3 The 15th term of an arithmetic sequence is 88. The sum of the first 10 terms is 310 .
Determine the first term and the common difference.