1.04j Sum to infinity: convergent geometric series |r|<1

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.

9

1. A geometric progression is such that the second term is equal to \(24 \%\) of the sum to infinity. Find the possible values of the common ratio.
2. An arithmetic progression \(P\) has first term \(a\) and common difference \(d\). An arithmetic progression \(Q\) has first term 2( \(a + 1\) ) and common difference ( \(d + 1\) ). It is given that $$\frac { 5 \text { th term of } P } { 12 \text { th term of } Q } = \frac { 1 } { 3 } \quad \text { and } \quad \frac { \text { Sum of first } 5 \text { terms of } P } { \text { Sum of first } 5 \text { terms of } Q } = \frac { 2 } { 3 } .$$ Find the value of \(a\) and the value of \(d\).

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

6 The first three terms of an arithmetic progression are \(\frac { p ^ { 2 } } { 6 } , 2 p - 6\) and \(p\).

1. Given that the common difference of the progression is not zero, find the value of \(p\).
2. Using this value, find the sum to infinity of the geometric progression with first two terms \(\frac { p ^ { 2 } } { 6 }\) and \(2 p - 6\).

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.

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.

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.

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.

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.

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.

7

1. Find the sum to infinity of the geometric progression with first three terms \(0.5,0.5 ^ { 3 }\) and \(0.5 ^ { 5 }\).
2. The first two terms in an arithmetic progression are 5 and 9. The last term in the progression is the only term which is greater than 200 . Find the sum of all the terms in the progression.

1 The first term of a geometric progression is 12 and the second term is - 6 . Find

1. the tenth term of the progression,
2. the sum to infinity.

8 A television quiz show takes place every day. On day 1 the prize money is \(\\) 1000$. If this is not won the prize money is increased for day 2 . The prize money is increased in a similar way every day until it is won. The television company considered the following two different models for increasing the prize money. Model 1: Increase the prize money by \(\\) 1000$ each day.
Model 2: Increase the prize money by \(10 \%\) each day.
On each day that the prize money is not won the television company makes a donation to charity. The amount donated is \(5 \%\) of the value of the prize on that day. After 40 days the prize money has still not been won. Calculate the total amount donated to charity

1. if Model 1 is used,
2. if Model 2 is used.

6

1. A geometric progression has a third term of 20 and a sum to infinity which is three times the first term. Find the first term.
2. An arithmetic progression is such that the eighth term is three times the third term. Show that the sum of the first eight terms is four times the sum of the first four terms.

7

1. The first two terms of an arithmetic progression are 1 and \(\cos ^ { 2 } x\) respectively. Show that the sum of the first ten terms can be expressed in the form \(a - b \sin ^ { 2 } x\), where \(a\) and \(b\) are constants to be found.
2. The first two terms of a geometric progression are 1 and \(\frac { 1 } { 3 } \tan ^ { 2 } \theta\) respectively, where \(0 < \theta < \frac { 1 } { 2 } \pi\).
   1. Find the set of values of \(\theta\) for which the progression is convergent.
   2. Find the exact value of the sum to infinity when \(\theta = \frac { 1 } { 6 } \pi\).