Mixed arithmetic and geometric

8 A progression has first term \(a\) and second term \(\frac { a ^ { 2 } } { a + 2 }\), where \(a\) is a positive constant.

1. For the case where the progression is geometric and the sum to infinity is 264 , find the value of \(a\).
2. For the case where the progression is arithmetic and \(a = 6\), determine the least value of \(n\) required for the sum of the first \(n\) terms to be less than - 480 .

4 The first term of a geometric progression and the first term of an arithmetic progression are both equal to \(a\). The third term of the geometric progression is equal to the second term of the arithmetic progression.
The fifth term of the geometric progression is equal to the sixth term of the arithmetic progression.
Given that the terms are all positive and not all equal, find the sum of the first twenty terms of the arithmetic progression in terms of \(a\).

4 The first term of an arithmetic progression is \(a\) and the common difference is - 4 . The first term of a geometric progression is \(5 a\) and the common ratio is \(- \frac { 1 } { 4 }\). The sum to infinity of the geometric progression is equal to the sum of the first eight terms of the arithmetic progression.

1. Find the value of \(a\).
   The \(k\) th term of the arithmetic progression is zero.
2. Find the value of \(k\).

9 The first term of a geometric progression is 216 and the fourth term is 64.

1. Find the sum to infinity of the progression.
   The second term of the geometric progression is equal to the second term of an arithmetic progression.
   The third term of the geometric progression is equal to the fifth term of the same arithmetic progression.
2. Find the sum of the first 21 terms of the arithmetic progression. \includegraphics[max width=\textwidth, alt={}, center]{8eb3d21b-dc45-493c-9e5c-3c0535c505e8-14_798_786_269_667} The diagram shows the circle \(x ^ { 2 } + y ^ { 2 } = 2\) and the straight line \(y = 2 x - 1\) intersecting at the points \(A\) and \(B\). The point \(D\) on the \(x\)-axis is such that \(A D\) is perpendicular to the \(x\)-axis.

4 A progression has a first term of 12 and a fifth term of 18.

1. Find the sum of the first 25 terms if the progression is arithmetic.
2. Find the 13th term if the progression is geometric.

6 The first term of an arithmetic progression is 12 and the sum of the first 9 terms is 135 .

1. Find the common difference of the progression. The first term, the ninth term and the \(n\)th term of this arithmetic progression are the first term, the second term and the third term respectively of a geometric progression.
2. Find the common ratio of the geometric progression and the value of \(n\).

10

1. The first and last terms of an arithmetic progression are 12 and 48 respectively. The sum of the first four terms is 57. Find the number of terms in the progression.
2. The third term of a geometric progression is four times the first term. The sum of the first six terms is \(k\) times the first term. Find the possible values of \(k\).

2 The first term in a progression is 36 and the second term is 32 .

1. Given that the progression is geometric, find the sum to infinity.
2. Given instead that the progression is arithmetic, find the number of terms in the progression if the sum of all the terms is 0 .

4 The 1st, 3rd and 13th terms of an arithmetic progression are also the 1st, 2nd and 3rd terms respectively of a geometric progression. The first term of each progression is 3 . Find the common difference of the arithmetic progression and the common ratio of the geometric progression.

5 The first three terms of an arithmetic progression are \(4 , x\) and \(y\) respectively. The first three terms of a geometric progression are \(x , y\) and 18 respectively. It is given that both \(x\) and \(y\) are positive.

1. Find the value of \(x\) and the value of \(y\).
2. Find the fourth term of each progression. \includegraphics[max width=\textwidth, alt={}, center]{2d5f452d-f820-40fc-9e22-9d3ac4f0698b-08_389_716_260_712} The diagram shows a triangle \(A B C\) in which \(B C = 20 \mathrm {~cm}\) and angle \(A B C = 90 ^ { \circ }\). The perpendicular from \(B\) to \(A C\) meets \(A C\) at \(D\) and \(A D = 9 \mathrm {~cm}\). Angle \(B C A = \theta ^ { \circ }\).
3. By expressing the length of \(B D\) in terms of \(\theta\) in each of the triangles \(A B D\) and \(D B C\), show that \(20 \sin ^ { 2 } \theta = 9 \cos \theta\).
4. Hence, showing all necessary working, calculate \(\theta\).

9. The first two terms of a geometric progression are 2 and \(x\) respectively, where \(x \neq 2\).

1. Find an expression for the third term in terms of \(x\). The first and third terms of arithmetic progression are 2 and \(x\) respectively.
2. Find an expression for the 11th term in terms of \(x\). Given that the third term of the geometric progression and the 11th term of the arithmetic progression have the same value,
3. find the value of \(x\),
4. find the sum of the first 50 terms of the arithmetic progression.

11 The curve \(y = \mathrm { f } ( x )\) is defined by the function \(\mathrm { f } ( x ) = \mathrm { e } ^ { - x } \sin x\) with domain \(0 \leq x \leq 4 \pi\).

1. 1. Show that the \(x\)-coordinates of the stationary points of the curve \(y = \mathrm { f } ( x )\), when arranged in increasing order, form an arithmetic sequence.
   2. Show that the corresponding \(y\)-coordinates form a geometric sequence.
2. Would the result still hold with a larger domain? Give reasons for your answer.