Shared terms between AP and GP

8 The first, second and third terms of an arithmetic progression are \(a , \frac { 3 } { 2 } a\) and \(b\) respectively, where \(a\) and \(b\) are positive constants. The first, second and third terms of a geometric progression are \(a , 18\) and \(b + 3\) respectively.

1. Find the values of \(a\) and \(b\).
2. Find the sum of the first 20 terms of the arithmetic progression.

6 A geometric progression has 6 terms. The first term is 192 and the common ratio is 1.5. An arithmetic progression has 21 terms and common difference 1.5. Given that the sum of all the terms in the geometric progression is equal to the sum of all the terms in the arithmetic progression, find the first term and the last term of the arithmetic progression.

7 The second term of a geometric progression is 3 and the sum to infinity is 12 .

1. Find the first term of the progression. An arithmetic progression has the same first and second terms as the geometric progression.
2. Find the sum of the first 20 terms of the arithmetic progression.

7 The first term of a geometric progression is 81 and the fourth term is 24 . Find

1. the common ratio of the progression,
2. the sum to infinity of the progression. The second and third terms of this geometric progression are the first and fourth terms respectively of an arithmetic progression.
3. Find the sum of the first ten terms of the arithmetic progression.

9

1. In an arithmetic progression, the sum, \(S _ { n }\), of the first \(n\) terms is given by \(S _ { n } = 2 n ^ { 2 } + 8 n\). Find the first term and the common difference of the progression.
2. The first 2 terms of a geometric progression are 64 and 48 respectively. The first 3 terms of the geometric progression are also the 1st term, the 9th term and the \(n\)th term respectively of an arithmetic progression. Find the value of \(n\).

6 The 1st, 2nd and 3rd terms of a geometric progression are the 1st, 9th and 21st terms respectively of an arithmetic progression. The 1st term of each progression is 8 and the common ratio of the geometric progression is \(r\), where \(r \neq 1\). Find

1. the value of \(r\),
2. the 4th term of each progression.

4 The 1st term of an arithmetic progression is \(a\) and the common difference is \(d\), where \(d \neq 0\).

1. Write down expressions, in terms of \(a\) and \(d\), for the 5th term and the 15th term. The 1st term, the 5th term and the 15th term of the arithmetic progression are the first three terms of a geometric progression.
2. Show that \(3 a = 8 d\).
3. Find the common ratio of the geometric progression.

8 The first term of an arithmetic progression is 8 and the common difference is \(d\), where \(d \neq 0\). The first term, the fifth term and the eighth term of this arithmetic progression are the first term, the second term and the third term, respectively, of a geometric progression whose common ratio is \(r\).

1. Write down two equations connecting \(d\) and \(r\). Hence show that \(r = \frac { 3 } { 4 }\) and find the value of \(d\).
2. Find the sum to infinity of the geometric progression.
3. Find the sum of the first 8 terms of the arithmetic progression.

5

1. An infinite geometric series has common ratio \(r\).
   The first term of the series is 10 and its sum to infinity is 50 .
   1. Show that \(r = \frac { 4 } { 5 }\).
   2. Find the second term of the series.
2. The first and second terms of the geometric series in part (a) have the same values as the 4th and 8th terms respectively of an arithmetic series.
   1. Find the common difference of the arithmetic series.
   2. The \(n\)th term of the arithmetic series is \(u _ { n }\). Find the value of \(\sum _ { n = 1 } ^ { 40 } u _ { n }\).

11 An arithmetic progression has first term \(a\) and common difference \(d\). The first, ninth and fourteenth terms are, respectively, the first three terms of a geometric progression with common ratio \(r\), where \(r \neq 1\).

1. Find \(d\) in terms of \(a\) and show that \(r = \frac { 5 } { 8 }\).
2. Find the sum to infinity of the geometric progression in terms of \(a\).

10 An arithmetic sequence and a geometric sequence have \(n\)th terms \(a _ { n }\) and \(g _ { n }\) respectively, where \(n = 1,2,3 , \ldots\). It is given that \(a _ { 1 } = g _ { 1 } , a _ { 2 } = g _ { 2 } , a _ { 5 } = g _ { 3 } , a _ { 1 } \neq a _ { 2 }\) and \(a _ { 1 } \neq 0\).

1. Show that the common ratio of the geometric sequence is 3 .
2. Find the common difference of the arithmetic sequence in terms of \(a _ { 1 }\).
3. Let \(a _ { 1 } = g _ { 1 } = 5\).
   1. Find the first three terms of both sequences.
   2. Show that every term of the geometric sequence is also a term of the arithmetic sequence.

11 An arithmetic progression has first term \(a\) and common difference \(d\). The first, ninth and fourteenth terms are, respectively, the first three terms of a geometric progression with common ratio \(r\), where \(r \neq 1\).

1. Find \(d\) in terms of \(a\) and show that \(r = \frac { 5 } { 8 }\).
2. Find the sum to infinity of the geometric progression in terms of \(a\).

An arithmetic progression has first term 5 and common difference \(d\), where \(d > 0\). The second, fifth and eleventh terms of the arithmetic progression, in that order, are the first three terms of a geometric progression.

1. Find the value of \(d\). [3]
2. The sum of the first 77 terms of the arithmetic progression is denoted by \(S_{77}\). The sum of the first 10 terms of the geometric progression is denoted by \(G_{10}\). Find the value of \(S_{77} - G_{10}\). [5]

An arithmetic progression (AP) and a geometric progression (GP) have the same first and fourth terms as each other. The first term of both is 1.5 and the fourth term of both is 12. Calculate the difference between the tenth terms of the AP and the GP. [5]

The positive integers \(x\), \(y\) and \(z\) are the first, second and third terms, respectively, of an arithmetic progression with common difference \(-4\). Also, \(x\), \(\frac{15}{y}\) and \(z\) are the first, second and third terms, respectively, of a geometric progression.

1. Show that \(y\) satisfies the equation \(y^4 - 16y^2 - 225 = 0\). [4]
2. Hence determine the sum to infinity of the geometric progression. [4]

An arithmetic progression has first term \(a\) and common difference \(d\), where \(a\) and \(d\) are non-zero. The first, third and fourth terms of the arithmetic progression are consecutive terms of a geometric progression with common ratio \(r\).

1. 1. Show that \(r = \frac{a + 2d}{a}\). [1]
   2. Find \(d\) in terms of \(a\). [2]
2. Find the common ratio of the geometric progression. [2]

An arithmetic progression has first term \(a\) and common difference \(d\). The first, ninth and fourteenth terms are, respectively, the first three terms of a geometric progression with common ratio \(r\), where \(r \neq 1\).

1. Find \(d\) in terms of \(a\) and show that \(r = \frac{5}{3}\). [7]
2. Find the sum to infinity of the geometric progression in terms of \(a\). [2]