Find n given sum condition

4 The sum, \(S _ { n }\), of the first \(n\) terms of an arithmetic progression is given by $$S _ { n } = n ^ { 2 } + 4 n$$ The \(k\) th term in the progression is greater than 200.
Find the smallest possible value of \(k\).

4 The first term of an arithmetic progression is 84 and the common difference is - 3 .

1. Find the smallest value of \(n\) for which the \(n\)th term is negative.
   It is given that the sum of the first \(2 k\) terms of this progression is equal to the sum of the first \(k\) terms.
2. Find the value of \(k\).

2 The first term of an arithmetic progression is - 20 and the common difference is 5 .

1. Find the sum of the first 20 terms of the progression.
   It is given that the sum of the first \(2 k\) terms is 10 times the sum of the first \(k\) terms.
2. Find the value of \(k\).

3 The first term of an arithmetic progression is 6 and the fifth term is 12 . The progression has \(n\) terms and the sum of all the terms is 90 . Find the value of \(n\).

1 The first term of an arithmetic progression is 61 and the second term is 57. The sum of the first \(n\) terms in \(n\). Find the value of the positive integer \(n\).

1 An arithmetic progression has first term - 12 and common difference 6 . The sum of the first \(n\) terms exceeds 3000 . Calculate the least possible value of \(n\).

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

In an arithmetic series,

• the sixth term is 2
• the sum of the first ten terms is - 80

For this series,

1. find the value of the first term and the value of the common difference.
2. Hence find the smallest value of \(n\) for which $$S _ { n } > 8000$$

4 The first 3 terms of an arithmetical progression are 7, 5.9 and 4.8.
Find

1. the common difference,
2. the smallest value of \(n\) for which the sum to \(n\) terms is negative.

8 The 25th term of an arithmetic series is 38 .
The sum of the first 40 terms of the series is 1250 .

1. Show that the common difference of this series is 1.5 .
2. Find the number of terms in the series which are less than 100 .

4 An arithmetic series has first term \(a\) and common difference \(d\).
The sum of the first 31 terms of the series is 310 .

1. Show that \(a + 15 d = 10\).
2. Given also that the 21st term is twice the 16th term, find the value of \(d\).
3. The \(n\)th term of the series is \(u _ { n }\). Given that \(\sum _ { n = 1 } ^ { k } u _ { n } = 0\), find the value of \(k\).

6 An arithmetic series has first term \(a\) and common difference \(d\). The sum of the first 25 terms of the series is 3500 .

1. Show that \(a + 12 d = 140\).
2. The fifth term of this series is 100 . Find the value of \(d\) and the value of \(a\).
3. The \(n\)th term of this series is \(u _ { n }\). Given that $$33 \left( \sum _ { n = 1 } ^ { 25 } u _ { n } - \sum _ { n = 1 } ^ { k } u _ { n } \right) = 67 \sum _ { n = 1 } ^ { k } u _ { n }$$ find the value of \(\sum _ { n = 1 } ^ { k } u _ { n }\).
   (3 marks)

4 The first three terms of an arithmetic series are \(9 p , 8 p - 3,5 p\) respectively, where \(p\) is a constant.
Given that the sum of the first \(n\) terms of this series is - 1512 , find the value of \(n\).

4 The positive integer \(k\) is such that $$\sum _ { r = 1 } ^ { k } ( 3 r - k ) = 90$$ Find the value of \(k\).