Arithmetic progression with parameters

4 The first, second and third terms of an arithmetic progression are \(k , 6 k\) and \(k + 6\) respectively.

1. Find the value of the constant \(k\).
2. Find the sum of the first 30 terms of the progression.

2 The first, second and third terms of an arithmetic progression are \(a , 2 a\) and \(a ^ { 2 }\) respectively, where \(a\) is a positive constant. Find the sum of the first 50 terms of the progression.

5 An arithmetic progression has first term \(a\) and common difference \(d\). It is given that the sum of the first 200 terms is 4 times the sum of the first 100 terms.

1. Find \(d\) in terms of \(a\).
2. Find the 100th term in terms of \(a\).

7

1. A geometric progression has first term \(a ( a \neq 0 )\), common ratio \(r\) and sum to infinity \(S\). A second geometric progression has first term \(a\), common ratio \(2 r\) and sum to infinity \(3 S\). Find the value of \(r\).
2. An arithmetic progression has first term 7. The \(n\)th term is 84 and the ( \(3 n\) )th term is 245 . Find the value of \(n\).

11. The first three terms of an arithmetic series are \(60,4 p\) and \(2 p - 6\) respectively.

1. Show that \(p = 9\)
2. Find the value of the 20th term of this series.
3. Prove that the sum of the first \(n\) terms of this series is given by the expression $$12 n ( 6 - n )$$ \includegraphics[max width=\textwidth, alt={}, center]{e878227b-d625-4ef2-ac49-a9dc05c5321a-27_106_68_2615_1877}

4. 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 $$a + 4 d = 6$$ Given also that the 8th term is half the 7th term,
2. find the values of \(a\) and \(d\).

14. The 5 th term of an arithmetic series is \(4 k\), where \(k\) is a constant. The sum of the first 8 terms of this series is \(20 k + 16\)

1. 1. Find, in terms of \(k\), an expression for the common difference of the series.
   2. Show that the first term of the series is \(16 - 8 k\) Given that the 9th term of the series is 24, find
2. the value of \(k\),
3. the sum of the first 20 terms. \includegraphics[max width=\textwidth, alt={}, center]{de511cb3-35c7-4225-b459-a136b6304b78-40_2257_54_314_1977}

1. In a large theatre there are 20 rows of seats.

The number of seats in the first row is \(a\), where \(a\) is a constant. In the second row the number of seats is \(( a + d )\), where \(d\) is a constant. In the third row the number of seats is \(( a + 2 d )\), and on each subsequent row there are \(d\) more seats than on the previous row. The number of seats in each row forms an arithmetic sequence. The total number of seats in the first 10 rows is 395

1. Use this information to show that \(10 a + 45 d = 395\) The total number of seats in the first 18 rows is 927
2. Use this information to write down a second simplified equation relating \(a\) and \(d\).
3. Solve these equations to find the value of \(a\) and the value of \(d\).
4. Find the number of seats in the 20th row of the theatre.

6. An arithmetic sequence has first term \(a\) and common difference \(d\). The sum of the first 10 terms of the sequence is 162 .

1. Show that \(10 a + 45 d = 162\) Given also that the sixth term of the sequence is 17 ,
2. write down a second equation in \(a\) and \(d\),
3. find the value of \(a\) and the value of \(d\).

1. In a large theatre there are \(n\) rows of seats, where \(n\) is a constant.

The number of seats in the first row is \(a\), where \(a\) is a constant.
In each subsequent row there are 4 more seats than in the previous row so that

• in the 2 nd row there are \(( a + 4 )\) seats
• in the 3rd row there are ( \(a + 8\) ) seats
• the number of seats in each row form an arithmetic sequence

Given that the total number of seats in the first 10 rows is 360

1. find the value of \(a\). Given also that the total number of seats in the \(n\) rows is 2146
2. show that $$n ^ { 2 } + 8 n - 1073 = 0$$
3. Hence
   1. state the number of rows of seats in the theatre,
   2. find the maximum number of seats in any one row.

6

1. The first three terms of an arithmetic progression are \(2 x , x + 4\) and \(2 x - 7\) respectively. Find the value of \(x\).
2. The first three terms of another sequence are also \(2 x , x + 4\) and \(2 x - 7\) respectively.
   (a) Verify that when \(x = 8\) the terms form a geometric progression and find the sum to infinity in this case.
   (b) Find the other possible value of \(x\) that also gives a geometric progression.

1. The second, third and fourth terms of an arithmetic sequence are \(2 k , 5 k - 10\) and \(7 k - 14\) respectively, where \(k\) is a constant.

Show that the sum of the first \(n\) terms of the sequence is a square number.

7. The first three terms of an arithmetic series are \(( 12 - p ) , 2 p\) and \(( 4 p - 5 )\) respectively, where \(p\) is a constant.

1. Find the value of \(p\).
2. Show that the sixth term of the series is 50 .
3. Find the sum of the first 15 terms of the series.
4. Find how many terms of the series have a value of less than 400.

9. The first two terms of an arithmetic series are \(( t - 1 )\) and \(\left( t ^ { 2 } - 5 \right)\) respectively, where \(t\) is a positive constant.

1. Find and simplify expressions in terms of \(t\) for
   1. the common difference of the series,
   2. the third term of the series. Given also that the third term of the series is 19 ,
2. find the value of \(t\),
3. show that the 10th term of the series is 75,
4. find the sum of the first 40 terms of the series.

6. 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.

3 An arithmetic progression has first term 2 and common difference \(d\), where \(d \neq 0\). The first, third and thirteenth terms of this progression are also the first, second and third terms, respectively, of a geometric progression. By determining \(d\), show that the arithmetic progression is an increasing sequence.

9 The first three terms of an arithmetic sequence are given by $$2 x + 5 \quad 5 x + 1 \quad 6 x + 7$$ 9

1. Show that \(x = 5\) is the only value which gives an arithmetic sequence.
   9
2. 1. Write down the value of the first term of the sequence.
      9
3. (ii) Find the value of the common difference of the sequence.
   9
4. The sum of the first \(N\) terms of the arithmetic sequence is \(S _ { N }\) where $$\begin{array} { r } S _ { N } < 100000 \\ S _ { N + 1 } > 100000 \end{array}$$ Find the value of \(N\).
   [0pt] [4 marks]