1.04h Arithmetic sequences: nth term and sum formulae

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 \(\theta = \frac { 1 } { 3 } \pi\).

2 The sum of the first 20 terms of an arithmetic progression is 405 and the sum of the first 40 terms is 1410 . Find the 60th term 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.

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 thirteenth term of an arithmetic progression is 12 and the sum of the first 30 terms is - 15 .
Find the sum of the first 50 terms of the progression.

4 The first, second and third terms of an arithmetic progression are \(k , 6 k\) and \(k + 6\) respectively.

1. Find the value of the constant \(k\).
2. Find the sum of the first 30 terms of the progression.

3 An arithmetic progression has first term 4 and common difference \(d\). The sum of the first \(n\) terms of the progression is 5863.

1. Show that \(( n - 1 ) d = \frac { 11726 } { n } - 8\).
2. Given that the \(n\)th term is 139 , find the values of \(n\) and \(d\), giving the value of \(d\) as a fraction.

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 .

7 The first term of an arithmetic progression is 1.5 and the sum of the first ten terms is 127.5 .

1. Find the common difference.
2. Find the sum of all the terms of the arithmetic progression whose values are between 25 and 100 .

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 \(S _ { E }\).

4 The sum, \(S _ { n }\), of the first \(n\) terms of an arithmetic progression is given by $$S _ { n } = n ^ { 2 } + 4 n$$ The \(k\) th term in the progression is greater than 200.
Find the smallest possible value of \(k\).

7 The first and second terms of an arithmetic progression are \(\frac { 1 } { \cos ^ { 2 } \theta }\) and \(- \frac { \tan ^ { 2 } \theta } { \cos ^ { 2 } \theta }\), respectively, where \(0 < \theta < \frac { 1 } { 2 } \pi\).

1. Show that the common difference is \(- \frac { 1 } { \cos ^ { 4 } \theta }\).
2. Find the exact value of the 13th term when \(\theta = \frac { 1 } { 6 } \pi\).

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 \(k\).

5 The first, third and fifth terms of an arithmetic progression are \(2 \cos x , - 6 \sqrt { 3 } \sin x\) and \(10 \cos x\) respectively, where \(\frac { 1 } { 2 } \pi < x < \pi\).

1. Find the exact value of \(x\).
2. Hence find the exact sum of the first 25 terms of the progression.

4 The first term of an arithmetic progression is 84 and the common difference is - 3 .

1. Find the smallest value of \(n\) for which the \(n\)th term is negative.
   It is given that the sum of the first \(2 k\) terms of this progression is equal to the sum of the first \(k\) terms.
2. Find the value of \(k\).

2 The first, second and third terms of an arithmetic progression are \(a , 2 a\) and \(a ^ { 2 }\) respectively, where \(a\) is a positive constant. Find the sum of the first 50 terms of the progression.

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.

3 An arithmetic progression has first term 7. The \(n\)th term is 84 and the ( \(3 n\) )th term is 245 .
Find the value of \(n\).

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.

4 In an arithmetic progression, the 1 st term is - 10 , the 15th term is 11 and the last term is 41 . Find the sum of all the terms in the 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.