1.04i Geometric sequences: nth term and finite series sum

8 The first term of a progression is \(\sin ^ { 2 } \theta\), where \(0 < \theta < \frac { 1 } { 2 } \pi\). The second term of the progression is \(\sin ^ { 2 } \theta \cos ^ { 2 } \theta\).

1. Given that the progression is geometric, find the sum to infinity.
   It is now given instead that the progression is arithmetic.
2. 1. Find the common difference of the progression in terms of \(\sin \theta\).
   2. Find the sum of the first 16 terms when \(\theta = \frac { 1 } { 3 } \pi\).

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.

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.

2 The second and third terms of a geometric progression are 10 and 8 respectively.
Find the sum to infinity.

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 .

10 The geometric progression \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) has first term 2 and common ratio \(r\) where \(r > 0\). It is given that \(\frac { 9 } { 2 } a _ { 5 } + 7 a _ { 3 } = 8\).

1. Find the value of \(r\).
2. Find the sum of the first 20 terms of the geometric progression. Give your answer correct to 4 significant figures. \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-14_2725_42_134_2008}
3. Find the sum to infinity of the progression \(a _ { 2 } , a _ { 5 } , a _ { 8 } , \ldots\).

9 The first term of a progression is \(\cos \theta\), where \(0 < \theta < \frac { 1 } { 2 } \pi\).

1. For the case where the progression is geometric, the sum to infinity is \(\frac { 1 } { \cos \theta }\).
   1. Show that the second term is \(\cos \theta \sin ^ { 2 } \theta\).
   2. Find the sum of the first 12 terms when \(\theta = \frac { 1 } { 3 } \pi\), giving your answer correct to 4 significant figures.
2. For the case where the progression is arithmetic, the first two terms are again \(\cos \theta\) and \(\cos \theta \sin ^ { 2 } \theta\) respectively. Find the 85 th term when \(\theta = \frac { 1 } { 3 } \pi\). \includegraphics[max width=\textwidth, alt={}, center]{54f3f051-e124-470d-87b5-8e25c35248a9-16_547_421_264_863} The diagram shows a sector \(A B C\) which is part of a circle of radius \(a\). The points \(D\) and \(E\) lie on \(A B\) and \(A C\) respectively and are such that \(A D = A E = k a\), where \(k < 1\). The line \(D E\) divides the sector into two regions which are equal in area.

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 circumference round the trunk of a large tree is measured and found to be 5.00 m . After one year the circumference is measured again and found to be 5.02 m .

1. Given that the circumferences at yearly intervals form an arithmetic progression, find the circumference 20 years after the first measurement.
2. Given instead that the circumferences at yearly intervals form a geometric progression, find the circumference 20 years after the first measurement.

8

1. An arithmetic progression is such that its first term is 6 and its tenth term is 19.5 .
   Find the sum of the first 100 terms of this arithmetic progression.
2. A geometric progression \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is such that \(a _ { 1 } = 24\) and the common ratio is \(\frac { 1 } { 2 }\). The sum to infinity of this geometric progression is denoted by \(S\). The sum to infinity of the even-numbered terms (i.e. \(a _ { 2 } , a _ { 4 } , a _ { 6 } , \ldots\) ) is denoted by \(S _ { E }\). Find the values of \(S\) and \(S _ { E }\).

8 A geometric progression has first term \(a\), common ratio \(r\) and sum to infinity \(S\). A second geometric progression has first term \(a\), common ratio \(R\) and sum to infinity \(2 S\).

1. Show that \(r = 2 R - 1\).
   It is now given that the 3rd term of the first progression is equal to the 2nd term of the second progression.
2. Express \(S\) in terms of \(a\).

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 In the expansion of \(( a + b x ) ^ { 7 }\), where \(a\) and \(b\) are non-zero constants, the coefficients of \(x , x ^ { 2 }\) and \(x ^ { 4 }\) are the first, second and third terms respectively of a geometric progression. Find the value of \(\frac { a } { b }\).

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 \(k\).

6 The second term of a geometric progression is 54 and the sum to infinity of the progression is 243 . The common ratio is greater than \(\frac { 1 } { 2 }\). Find the tenth term, giving your answer in exact form.

7 A tool for putting fence posts into the ground is called a 'post-rammer'. The distances in millimetres that the post sinks into the ground on each impact of the post-rammer follow a geometric progression. The first three impacts cause the post to sink into the ground by \(50 \mathrm {~mm} , 40 \mathrm {~mm}\) and 32 mm respectively.

1. Verify that the 9th impact is the first in which the post sinks less than 10 mm into the ground.
2. Find, to the nearest millimetre, the total depth of the post in the ground after 20 impacts.
3. Find the greatest total depth in the ground which could theoretically be achieved.

4 A geometric progression is such that the third term is 1764 and the sum of the second and third terms is 3444 . Find the 50th term.

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.

7 The sum of the first two terms of a geometric progression is 15 and the sum to infinity is \(\frac { 125 } { 7 }\). The common ratio of the progression is negative. Find the third term 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.

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.

1 A geometric progression has first term 64 and sum to infinity 256. Find

1. the common ratio,
2. the sum of the first ten terms.

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.

3 Each year a company gives a grant to a charity. The amount given each year increases by \(5 \%\) of its value in the preceding year. The grant in 2001 was \(\\) 5000$. Find

1. the grant given in 2011,
2. the total amount of money given to the charity during the years 2001 to 2011 inclusive.

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.