Largest or extreme value of sum

11. The first term of an arithmetic sequence is 30 and the common difference is - 1.5

1. Find the value of the 25th term. The \(r\) th term of the sequence is 0 .
2. Find the value of \(r\). The sum of the first \(n\) terms of the sequence is \(S _ { n }\).
3. Find the largest positive value of \(S _ { n }\).

1．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 \(S _ { n }\) ．

8 An arithmetic series has first term 9300 and 10th term 3900.

1. Show that the 20th term of the series is negative.
2. The sum of the first \(n\) terms is denoted by \(S\). Find the greatest value of \(S\) as \(n\) varies.

9. The third term of an arithmetic series is \(5 \frac { 1 } { 2 }\). The sum of the first four terms of the series is \(22 \frac { 3 } { 4 }\).

1. Show that the first term of the series is \(6 \frac { 1 } { 4 }\) and find the common difference.
2. Find the number of positive terms in the series.
3. Hence, find the greatest value of the sum of the first \(n\) terms of the series.

8 An arithmetic series has first term \(a\) and common difference \(d\). The sum of the first 5 terms of the series is 575 .

1. Show that \(a + 2 d = 115\).
2. Given also that the 10th term of the series is 87, find the value of \(d\).
3. The \(n\)th term of the series is \(u _ { n }\). Given that \(u _ { k } > 0\) and \(u _ { k + 1 } < 0\), find the value of \(\sum _ { n = 1 } ^ { k } u _ { n }\).
   [0pt] [5 marks]

5 An arithmetic sequence has first term \(a\) and common difference \(d\). The sum of the first 16 terms of the sequence is 260 5

1. Show that \(4 a + 30 d = 65\) 5
2. Given that the sum of the first 60 terms is 315 , find the sum of the first 41 terms.
   5
3. \(\quad S _ { n }\) 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 \(S _ { n }\)