Sum of first n terms

2 The thirteenth term of an arithmetic progression is 12 and the sum of the first 30 terms is - 15 .
Find the sum of the first 50 terms of the progression.

4 In an arithmetic progression, the 1 st term is - 10 , the 15th term is 11 and the last term is 41 . Find the sum of all the terms in the progression.

A girl saves money over a period of 200 weeks. She saves 5 p in Week 1,7 p in Week 2, 9p in Week 3, and so on until Week 200. Her weekly savings form an arithmetic sequence.

1. Find the amount she saves in Week 200.
2. Calculate her total savings over the complete 200 week period.

1 In an arithmetic progression the first term is 15 and the twentieth term is 72. Find the sum of the first 100 terms.

6 A sequence is given by $$\begin{gathered} a _ { 1 } = 4 \\ a _ { r + 1 } = a _ { r } + 3 \end{gathered}$$ Write down the first 4 terms of this sequence.
Find the sum of the first 100 terms of the sequence.

2 The \(n\)th term of an arithmetic progression is \(6 + 5 n\). Find the sum of the first 20 terms.

11

1. André is playing a game where he makes piles of counters. He puts 3 counters in the first pile. Each successive pile he makes has 2 more counters in it than the previous one.
   1. How many counters are there in his sixth pile?
   2. André makes ten piles of counters. How many counters has he used altogether?
2. In another game, played with an ordinary fair die and counters, Betty needs to throw a six to start. The probability \(\mathrm { P } _ { n }\) of Betty starting on her \(n\)th throw is given by $$P _ { n } = \frac { 1 } { 6 } \times \left( \frac { 5 } { 6 } \right) ^ { n - 1 }$$
   1. Calculate \(\mathrm { P } _ { 4 }\). Give your answer as a fraction.
   2. The values \(\mathrm { P } _ { 1 } , \mathrm { P } _ { 2 } , \mathrm { P } _ { 3 } , \ldots\) form an infinite geometric progression. State the first term and the common ratio of this progression. Hence show that \(\mathrm { P } _ { 1 } + \mathrm { P } _ { 2 } + \mathrm { P } _ { 3 } + \ldots = 1\).
   3. Given that \(\mathrm { P } _ { n } < 0.001\), show that \(n\) satisfies the inequality $$n > \frac { \log _ { 10 } 0.006 } { \log _ { 10 } \left( \frac { 5 } { 6 } \right) } + 1$$ Hence find the least value of \(n\) for which \(\mathrm { P } _ { n } < 0.001\).

1 The arithmetic series $$23 + 32 + 41 + 50 + \ldots + 2534$$ has 280 terms.

1. Write down the common difference of the series.
2. Find the 100th term of the series.
3. Find the sum of the 280 terms of the series.

10

1. An arithmetic sequence has 300 terms. The first term of the sequence is - 7 and the last term is 32 Find the sum of the 300 terms.
   [0pt] [2 marks]
   10
2. A school holds a raffle at its summer fair. There are nine prizes.
   The total value of the prizes is \(\pounds 1260\) The values of the prizes form an arithmetic sequence.
   The top prize has the highest value, and the bottom prize has the least value.
   The value of the top prize is six times the value of the bottom prize.
   Find the value of the top prize.