Form and solve quadratic in parameter

5 The fifth, sixth and seventh terms of a geometric progression are \(8 k , - 12\) and \(2 k\) respectively. Given that \(k\) is negative, find the sum to infinity of the progression.

2 The first, second and third terms of a geometric progression are \(2 p + 6 , - 2 p\) and \(p + 2\) respectively, where \(p\) is positive. Find the sum to infinity of the progression.

5 The first, second and third terms of a geometric progression are \(2 p + 6,5 p\) and \(8 p + 2\) respectively.

1. Find the possible values of the constant \(p\).
2. One of the values of \(p\) found in (a) is a negative fraction. Use this value of \(p\) to find the sum to infinity of this progression.

8 The first term of a progression is \(4 x\) and the second term is \(x ^ { 2 }\).

1. For the case where the progression is arithmetic with a common difference of 12 , find the possible values of \(x\) and the corresponding values of the third term.
2. For the case where the progression is geometric with a sum to infinity of 8 , find the third term.

8

1. Over a 21-day period an athlete prepares for a marathon by increasing the distance she runs each day by 1.2 km . On the first day she runs 13 km .
   1. Find the distance she runs on the last day of the 21-day period.
   2. Find the total distance she runs in the 21-day period.
2. The first, second and third terms of a geometric progression are \(x , x - 3\) and \(x - 5\) respectively.
   1. Find the value of \(x\).
   2. Find the fourth term of the progression.
   3. Find the sum to infinity of the progression.

9 The first, second and third terms of a geometric progression are \(3 k , 5 k - 6\) and \(6 k - 4\), respectively.

1. Show that \(k\) satisfies the equation \(7 k ^ { 2 } - 48 k + 36 = 0\).
2. Find, showing all necessary working, the exact values of the common ratio corresponding to each of the possible values of \(k\).
3. One of these ratios gives a progression which is convergent. Find the sum to infinity.

8 The first term of a progression is \(4 x\) and the second term is \(x ^ { 2 }\).

1. For the case where the progression is arithmetic with a common difference of 12 , find the possible values of \(x\) and the corresponding values of the third term.
2. For the case where the progression is geometric with a sum to infinity of 8 , find the third term.

16. The first three terms of a geometric series are \(( k + 5 ) , k\) and \(( 2 k - 24 )\) respectively, where \(k\) is a constant.

1. Show that \(k ^ { 2 } - 14 k - 120 = 0\)
2. Hence find the possible values of \(k\).
3. Given that the series is convergent, find
   1. the common ratio,
   2. the sum to infinity.

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      END

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

A software developer released an app to download.
The numbers of downloads of the app each month, in thousands, for the first three months after the app was released were $$2 k - 15 \quad k \quad k + 4$$ where \(k\) is a constant.
Given that the numbers of downloads each month are modelled as a geometric series,

1. show that \(k ^ { 2 } - 7 k - 60 = 0\)
2. predict the number of downloads in the 4th month. The total number of all downloads of the app is predicted to exceed 3 million for the first time in the \(N\) th month.
3. Calculate the value of \(N\) according to the model.

5. The first three terms of a geometric series are \(4 p , ( 3 p + 15 )\) and ( \(5 p + 20\) ) respectively, where \(p\) is a positive constant.

1. Show that \(11 p ^ { 2 } - 10 p - 225 = 0\)
2. Hence show that \(p = 5\)
3. Find the common ratio of this series.
4. Find the sum of the first ten terms of the series, giving your answer to the nearest integer.

9. The first three terms of a geometric sequence are $$7 k - 5,5 k - 7,2 k + 10$$ where \(k\) is a constant.

1. Show that \(11 k ^ { 2 } - 130 k + 99 = 0\) Given that \(k\) is not an integer,
2. show that \(k = \frac { 9 } { 11 }\) For this value of \(k\),
3. 1. evaluate the fourth term of the sequence, giving your answer as an exact fraction,
   2. evaluate the sum of the first ten terms of the sequence.

9 A geometric progression has first term \(a\) and common ratio \(r\), and the terms are all different. The first, second and fourth terms of the geometric progression form the first three terms of an arithmetic progression.

1. Show that \(r ^ { 3 } - 2 r + 1 = 0\).
2. Given that the geometric progression converges, find the exact value of \(r\).
3. Given also that the sum to infinity of this geometric progression is \(3 + \sqrt { 5 }\), find the value of the integer \(a\).

9 A geometric progression has a positive common ratio. Its first three terms are 32, \(b\) and 12.5.
Find the value of \(b\) and find also the sum of the first 15 terms of the progression.

1. The first three terms of a geometric sequence are

$$3 k + 4 \quad 12 - 3 k \quad k + 16$$ where \(k\) is a constant.

1. Show that \(k\) satisfies the equation $$3 k ^ { 2 } - 62 k + 40 = 0$$ Given that the sequence converges,
2. 1. find the value of \(k\), giving a reason for your answer,
   2. find the value of \(S _ { \infty }\)

1. The first 3 terms of a geometric sequence are

$$3 ^ { 4 k - 5 } \quad 9 ^ { 7 - 2 k } \quad 3 ^ { 2 ( k - 1 ) }$$ where \(k\) is a constant.

1. Using algebra and making your reasoning clear, prove that \(k = \frac { 5 } { 2 }\)
2. Hence find the sum to infinity of the geometric sequence.

11 The first three terms of a geometric sequence are \(5 k - 2,3 k - 6 , k + 2\), where \(k\) is a constant.

1. Show that \(k\) satisfies the equation \(k ^ { 2 } - 11 k + 10 = 0\).
2. When \(k\) takes the smaller of the two possible values, find the sum of the first 20 terms of the sequence.
3. When \(k\) takes the larger of the two possible values, find the sum to infinity of the sequence.

9. The first three terms of a geometric series are \(( x - 2 ) , ( x + 6 )\) and \(x ^ { 2 }\) respectively.

1. Show that \(x\) must be a solution of the equation $$x ^ { 3 } - 3 x ^ { 2 } - 12 x - 36 = 0$$
2. Verify that \(x = 6\) is a solution of equation (I) and show that there are no other real solutions. Using \(x = 6\),
3. find the common ratio of the series,
4. find the sum of the first eight terms of the series.

2. The first three terms of a geometric series are ( \(p - 1\) ), 2 and ( \(2 p + 5\) ) respectively, where \(p\) is a constant. Find the two possible values of \(p\).

9 It is given that \(x , 6\) and \(x + 5\) are consecutive terms of a geometric progression.

1. Show that \(x ^ { 2 } + 5 x - 36 = 0\) and find the possible values of \(x\).
2. Hence find the possible values of the common ratio. Furthermore, \(x , 6\) and \(x + 5\) are the second, third and fourth terms of a geometric progression for which the sum to infinity exists.
3. Find the first term and the sum to infinity.

6

1. Given that the numbers \(a , b\) and \(c\) are in arithmetic progression, show that \(a + c = 2 b\).
2. Find an analogous result for three numbers in geometric progression.
3. The numbers \(2 - 3 x , 2 x , 3 - 2 x\) are the first three terms of a convergent geometric progression. Find \(x\) and hence calculate the sum to infinity.

The first three terms of a geometric series are \((x - 2)\), \((x + 6)\) and \(x^2\) respectively.

1. Show that \(x\) must be a solution of the equation $$x^3 - 3x^2 - 12x - 36 = 0. \quad \text{(I)}$$ [3]
2. Verify that \(x = 6\) is a solution of equation (I) and show that there are no other real solutions. [6]

Using \(x = 6\),

1. find the common ratio of the series, [1]
2. find the sum of the first eight terms of the series. [2]

The first three terms of a geometric sequence are $$u_1 = 3k + 4 \quad u_2 = 12 - 3k \quad u_3 = k + 16$$ where \(k\) is a constant. Given that the sequence converges,

1. Find the value of \(k\), giving a reason for your answer. [4]
2. Find the value of \(\sum_{r=2}^{\infty} u_r\). [3]

The first three terms of a geometric sequence are $$u_1 = 3k + 4 \quad u_2 = 12 - 3k \quad u_3 = k + 16$$ where \(k\) is a constant. Given that the sequence converges,

1. Find the value of k, giving a reason for your answer. [4]
2. Find the value of \(\sum_{r=2}^{\infty} u_r\) [3]