1.04h Arithmetic sequences: nth term and sum formulae

1 The 7th term of an arithmetic progression is 6. The sum of the first 10 terms of the progression is 30. Find the 5th term of the progression.

3 A sequence is given by $$\begin{gathered} a _ { 1 } = 4 \\ a _ { r + 1 } = a _ { r } + 3 \end{gathered}$$ Write down the first 4 terms of this sequence.
Find the sum of the first 100 terms of the sequence.

2 The \(n\)th term of a sequence, \(u _ { n }\), is given by $$u _ { n } = 12 - \frac { 1 } { 2 } n .$$

1. Write down the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\). State what type of sequence this is.
2. Find \(\sum _ { n = 1 } ^ { 30 } u _ { n }\).

5 Jim and Mary are each planning monthly repayments for money they want to borrow.

1. Jim's first payment is \(\pounds 500\), and he plans to pay \(\pounds 10\) less each month, so that his second payment is \(\pounds 490\), his third is \(\pounds 480\), and so on.
   (A) Calculate his 12th payment.
   (B) He plans to make 24 payments altogether. Show that he pays \(\pounds 9240\) in total.
2. Mary's first payment is \(\pounds 460\) and she plans to pay \(2 \%\) less each month than the previous month, so that her second payment is \(\pounds 450.80\), her third is \(\pounds 441.784\), and so on.
   (A) Calculate her 12th payment.
   (B) Show that Jim's 20th payment is less than Mary's 20th payment but that his 19th is not less than her 19th.
   (C) Mary plans to make 24 payments altogether. Calculate how much she pays in total.
   (D) How much would Mary's first payment need to be if she wishes to pay \(2 \%\) less each month as before, but to pay the same in total as Jim, \(\pounds 9240\), over the 24 months?

3 A sequence begins $$\begin{array} { l l l l l l l l l l l l } 1 & 2 & 3 & 4 & 5 & 1 & 2 & 3 & 4 & 5 & 1 & \ldots \end{array}$$ and continues in this pattern.

1. Find the 48th term of this sequence.
2. Find the sum of the first 48 terms of this sequence.

5. \begin{figure}[h] \end{figure} Figure 1 shows part of a sequence \(S _ { 1 } , S _ { 2 } , S _ { 3 } , \ldots\), of model snowflakes. The first term \(S _ { 1 }\) consists of a single square of side \(a\). To obtain \(S _ { 2 }\), the middle third of each edge is replaced with a new square, of side \(\frac { a } { 3 }\), as shown in Figure 1 . Subsequent terms are obtained by replacing the middle third of each external edge of a new square formed in the previous snowflake, by a square \(\frac { 1 } { 3 }\) of the size, as illustrated by \(S _ { 3 }\) in Figure 1.

1. Deduce that to form \(S _ { 4 } , 36\) new squares of side \(\frac { a } { 27 }\) must be added to \(S _ { 3 }\).
2. Show that the perimeters of \(S _ { 2 }\) and \(S _ { 3 }\) are \(\frac { 20 a } { 3 }\) and \(\frac { 28 a } { 3 }\) respectively.
3. Find the perimeter of \(S _ { n }\).
4. Describe what happens to the perimeter of \(S _ { n }\) as \(n\) increases.
5. Find the areas of \(S _ { 1 } , S _ { 2 }\) and \(S _ { 3 }\).
6. Find the smallest value of the constant \(S\) such that the area of \(S _ { n } < S\), for all values of \(n\). \includegraphics[max width=\textwidth, alt={}, center]{f2290882-b9a4-43ec-a38f-c44d46477242-5_590_1041_283_588} Figure 2 shows a sketch of the curve \(C\) with equation \(y = \tan \frac { t } { 2 } , \quad 0 \leq t \leq \frac { \pi } { 2 }\).
   The point \(P\) on \(C\) has coordinates \(\left( x , \tan \frac { x } { 2 } \right)\).
   The vertices of rectangle \(R\) are at \(( x , 0 ) , \left( \frac { x } { 2 } , 0 \right) , \left( \frac { x } { 2 } , \tan \frac { x } { 2 } \right)\) and \(\left( x , \tan \frac { x } { 2 } \right)\) as shown in Figure 2.

5.(a)The sides of the triangle \(A B C\) have lengths \(B C = a , A C = b\) and \(A B = c\) ,where \(a < b < c\) .The sizes of the angles \(A , B\) and \(C\) form an arithmetic sequence.

1. Show that the area of triangle \(A B C\) is \(a c \frac { \sqrt { 3 } } { 4 }\) . Given that \(a = 2\) and \(\sin A = \frac { \sqrt { } 15 } { 5 }\) ,find
2. the value of \(b\) ,
3. the value of \(c\) .
   (b)The internal angles of an \(n\)-sided polygon form an arithmetic sequence with first term \(143 ^ { \circ }\) and common difference \(2 ^ { \circ }\) . Given that all of the internal angles are less than \(180 ^ { \circ }\) ,find the value of \(n\) .

2.The sum of the first \(p\) terms of an arithmetic series is \(q\) and the sum of the first \(q\) terms of the same arithmetic series is \(p\) ,where \(p\) and \(q\) are positive integers and \(p \neq q\) . Giving simplified answers in terms of \(p\) and \(q\) ,find

1. the common difference of the terms in this series,
2. the first term of the series,
3. the sum of the first \(( p + q )\) terms of the series.

4.A sequence of positive integers \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) has \(r\) th term given by $$a _ { r } = 2 ^ { r } - 1$$

1. Write down the first 6 terms of this sequence.
2. Verify that \(a _ { r + 1 } = 2 a _ { r } + 1\)
3. Find \(\sum _ { r = 1 } ^ { n } a _ { r }\)
4. Show that \(\frac { 1 } { a _ { r + 1 } } < \frac { 1 } { 2 } \times \frac { 1 } { a _ { r } }\)
5. Hence show that \(1 + \frac { 1 } { 3 } + \frac { 1 } { 7 } + \frac { 1 } { 15 } + \frac { 1 } { 31 } + \ldots < 1 + \frac { 1 } { 3 } + \left( \frac { 1 } { 7 } + \frac { \frac { 1 } { 2 } } { 7 } + \frac { \frac { 1 } { 4 } } { 7 } + \ldots \right)\)
6. Show that \(\frac { 31 } { 21 } < \sum _ { r = 1 } ^ { \infty } \frac { 1 } { a _ { r } } < \frac { 34 } { 21 }\)

8. In the year 2000 a shop sold 150 computers. Each year the shop sold 10 more computers than the year before, so that the shop sold 160 computers in 2001, 170 computers in 2002, and so on forming an arithmetic sequence.

1. Show that the shop sold 220 computers in 2007.
2. Calculate the total number of computers the shop sold from 2000 to 2013 inclusive. In the year 2000, the selling price of each computer was \(\pounds 900\). The selling price fell by \(\pounds 20\) each year, so that in 2001 the selling price was \(\pounds 880\), in 2002 the selling price was \(\pounds 860\), and so on forming an arithmetic sequence.
3. In a particular year, the selling price of each computer in \(\pounds s\) was equal to three times the number of computers the shop sold in that year. By forming and solving an equation, find the year in which this occurred.

3 A sequence of terms \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { n } = 24 - \frac { 2 } { 3 } n$$

1. Write down the exact values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
2. Find the value of \(k\) such that \(u _ { k } = 0\).
3. Find \(\sum _ { n = 1 } ^ { 20 } u _ { n }\).

8 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 8 \quad \text { and } \quad u _ { n + 1 } = u _ { n } + 3 .$$

1. Show that \(u _ { 5 } = 20\).
2. The \(n\)th term of the sequence can be written in the form \(u _ { n } = p n + q\). State the values of \(p\) and \(q\).
3. State what type of sequence it is.
4. Find the value of \(N\) such that \(\sum _ { n = 1 } ^ { 2 N } u _ { n } - \sum _ { n = 1 } ^ { N } u _ { n } = 1256\).

2 A sequence \(S\) has terms \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) defined by \(u _ { n } = 3 n + 2\) for \(n \geqslant 1\).

1. Write down the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
2. State what type of sequence \(S\) is.
3. Find \(\sum _ { n = 101 } ^ { 200 } u _ { n }\).

6 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { n } = 85 - 5 n\) for \(n \geqslant 1\).

1. Write down the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
2. Find \(\sum _ { n = 1 } ^ { 20 } u _ { n }\).
3. Given that \(u _ { 1 } , u _ { 5 }\) and \(u _ { p }\) are, respectively, the first, second and third terms of a geometric progression, find the value of \(p\).
4. Find the sum to infinity of the geometric progression in part (iii).

2 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 7 \text { and } u _ { n + 1 } = u _ { n } + 4 \text { for } n \geqslant 1 .$$

1. Show that \(u _ { 17 } = 71\).
2. Show that \(\sum _ { n = 1 } ^ { 35 } u _ { n } = \sum _ { n = 36 } ^ { 50 } u _ { n }\).

6

1. The first three terms of an arithmetic progression are \(2 x , x + 4\) and \(2 x - 7\) respectively. Find the value of \(x\).
2. The first three terms of another sequence are also \(2 x , x + 4\) and \(2 x - 7\) respectively.
   1. Verify that when \(x = 8\) the terms form a geometric progression and find the sum to infinity in this case.
   2. Find the other possible value of \(x\) that also gives a geometric progression.

2 The tenth term of an arithmetic progression is equal to twice the fourth term. The twentieth term of the progression is 44 .

1. Find the first term and the common difference.
2. Find the sum of the first 50 terms.

7

1. The first term of a geometric progression is 7 and the common ratio is - 2 .
   1. Find the ninth term.
   2. Find the sum of the first 15 terms.
2. The first term of an arithmetic progression is 7 and the common difference is - 2 . The sum of the first \(N\) terms is - 2900 . Find the value of \(N\).

5

1. A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 4 \quad \text { and } \quad u _ { n + 1 } = \frac { 2 } { u _ { n } } \quad \text { for } n \geqslant 1 .$$
   1. Write down the values of \(u _ { 2 }\) and \(u _ { 3 }\).
   2. Describe the behaviour of the sequence.
2. In an arithmetic progression the ninth term is 18 and the sum of the first nine terms is 72. Find the first term and the common difference.

9

1. An arithmetic progression has first term \(\log _ { 2 } 27\) and common difference \(\log _ { 2 } x\).
   1. Show that the fourth term can be written as \(\log _ { 2 } \left( 27 x ^ { 3 } \right)\).
   2. Given that the fourth term is 6, find the exact value of \(x\).
2. A geometric progression has first term \(\log _ { 2 } 27\) and common ratio \(\log _ { 2 } y\).
   1. Find the set of values of \(y\) for which the geometric progression has a sum to infinity.
   2. Find the exact value of \(y\) for which the sum to infinity of the geometric progression is 3 . \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}

6 Sarah is carrying out a series of experiments which involve using increasing amounts of a chemical. In the first experiment she uses 6 g of the chemical and in the second experiment she uses 7.8 g of the chemical.

1. Given that the amounts of the chemical used form an arithmetic progression, find the total amount of chemical used in the first 30 experiments.
2. Instead it is given that the amounts of the chemical used form a geometric progression. Sarah has a total of 1800 g of the chemical available. Show that \(N\), the greatest number of experiments possible, satisfies the inequality $$1.3 ^ { N } \leqslant 91 ,$$ and use logarithms to calculate the value of \(N\).

7 In an arithmetic progression the first term is 5 and the common difference is 3 . The \(n\)th term of the progression is denoted by \(u _ { n }\).

1. Find the value of \(u _ { 20 }\).
2. Show that \(\sum _ { n = 10 } ^ { 20 } u _ { n } = 517\).
3. Find the value of \(N\) such that \(\sum _ { n = N } ^ { 2 N } u _ { n } = 2750\).

6 An arithmetic progression \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { 1 } = 5\) and \(u _ { n + 1 } = u _ { n } + 1.5\) for \(n \geqslant 1\).

1. Given that \(u _ { k } = 140\), find the value of \(k\). A geometric progression \(w _ { 1 } , w _ { 2 } , w _ { 3 } , \ldots\) is defined by \(w _ { n } = 120 \times ( 0.9 ) ^ { n - 1 }\) for \(n \geqslant 1\).
2. Find the sum of the first 16 terms of this geometric progression, giving your answer correct to 3 significant figures.
3. Use an algebraic method to find the smallest value of \(N\) such that \(\sum _ { n = 1 } ^ { N } u _ { n } > \sum _ { n = 1 } ^ { \infty } w _ { n }\).

6 An arithmetic progression has first term 7 and third term 12.

1. Find the 20th term of this progression.
2. Find the sum of the 21st to the 50th terms inclusive of this progression.

6 The third term of an arithmetic progression is 24 . The tenth term is 3 .
Find the first term and the common difference. Find also the sum of the 21st to 50th terms inclusive.