Sequence [CIE E math 2017-2018 Topic question]

1	(2017/w/22/q16) Here are the four terms of a sequence $23 \;\;\;\; 17\;\;\;\; 11\;\;\;\; 5$ (a)Find the next term (b)Find the $n$th term	1 2
2	(2018/s/22/q3) Here is a sequence. $ a,\;\;\;\; 13,\;\;\;\; 9,\;\;\;\; 3,\;\;\;\; -5,\;\;\;\; -1,\;\;\;\; b,\;\;\;\; ...$ Find the value of $a$ and $b$	2
3	(2018/s/22/q5) $22\;\;\;\; 17\;\;\;\; 25\;\;\;\; 41\;\;\;\; 39\;\;\;\; 4\;\;\;\;$ Work out the difference between the two prime numbers in the list above.	2
4	(2018/w/22/q13) These are the first five terms of a sequence. $-4\;\;\;\; 2\;\;\;\; 8\;\;\;\; 14\;\;\;\; 20\;\;\;\;$ Find the expression of the $n$th term of a sequence.	2
5	(2017/s/41/q9) (a)The $n$th term of a sequence is $8n-3$. (i)Write down the first two terms of this sequence. (ii)Show that the number $203$ is not in this sequence. (b)Find the $n$th term of these sequences. (i) $13,\;\;\;\;19,\;\;\;\;25,\;\;\;\;31,\;\;\;\;...$ (ii)  $4,\;\;\;\;8,\;\;\;\;14,\;\;\;\;22,\;\;\;\;....$ (c)  $...,\;\;\;\;20,\;\;\;\;50,\;\;\;\;...$ The second term of this sequence is 20 and the third term is 50. The rule for finding the next term in this sequence is subtract y then multiply by 5. Find the value of y and work out the first term of this sequence	1 2 2 2 4
6	(2017/w/41/q6) These are the first four diagrams in a sequence. Each diagram is made from small squares and crosses. (a)Complete the table. (b)Find the number of crosses in Diagram 60. (c) Which diagram has 226 squares? (d) The side of each small square has length 1cm. The number of lines of length 1cm in Diagram $n$ is $2n^2 + 2n + q$. Find the value of $q$.	6 1 1 2
7	(2018/s/41/q12) Marco is making patterns with grey and white circular mats. The patterns form a sequence. Marco makes a table to show some information about the patterns. (a) Complete the table for Pattern $5$. (b) Find an expression, in terms of $n$, for the number of grey mats in Pattern $n$. (c) Marco makes a pattern with $24$ grey mats. Find the total number of mats in this pattern. (d) Marco needs a total of $6$ mats to make the first pattern. He needs a total of $16$ mats to make the first two patterns. He needs a total of $\frac16 n^3+an^2+bn$ mats to make the first $n$ patterns. Find the value of $a$ and the value of $b$.	2 2 2 6
8	(2017/m/42/q11) On Monday, Ankuri sent this text message to two friends. Today is Day Number 1. Tomorrow, please add 1 to the Day Number and send this text message to two friends. All the friends who receive a text message follow the instructions. (a) Complete the table (b) Write down an expression for the number of text messages sent on Day Number $n$ (c) Ankuri thinks that, by the end of Day Number 3, the total number of text messages that have been sent is $2^4-2$. (i) Show that she is correct. (ii) Complete the statement. The total number of text messages sent by the end of Day Number 5 is ........................ which is equal to $2^k – 2$ where k = ........................ . (iii) Write down an expression for the total number of text messages sent by the end of Day Number $n$. (iv) Find the Day Number when the total number of text messages sent by the end of the day is 1022.	4 1 2 2 1 1

Short Answer 

1. (a) -1 (b) \( -6 n+29 \) 
2. \( a=15, b=-27 \) 
3. 24 
4. \( 6 n-10 \) 
5. (a)(i) 5 and 13 (a)(ii) \( 8 n-3=203 ; 25.75 \) or \( 25 \frac{3}{4} \)
    (b)(i) \( 6 n+7 \) (b)(ii) \( n^{2}+n+2 \)
    (c) \( [y=] 10[ \) First term \( =] 14 \)

6. (a) \( 18,22,4 n+2 ; 17,26, n^{2}+1 \) (b) 242 (c) 15 (d) 3
7. (a) 18,28 (b) \( 3 n+3 \) oe (c) 45 (d) \( a+\frac{3}{2}, b=\frac{13}{3} \)
8. (a) 4  5  6  7 and 8  16  32  6  128   (b) \( 2^{n} \) (c) (i) shown (ii) 62 and 6 (iii) \( 2^{n+1}-2 \) (iv) 9
9. (a)25 9 16 (b)(i)$(n-1)^2$ (ii) n+3 (c)25 (d)(i)$n^2-3n-2$ (ii)808
10. \( 64 x(n+3)^{2} ; 17,3 n+2 ; 47,(n+3)^{2}-(3 n+2) \); $\frac{6}{7}, \frac{n+2}{n+1}$

11. (a) \( \frac{8}{15}, \frac{n+2}{2 n+3} \)
(b) (i) \( 1-2 n \)
(b)(ii) \( n^{3}+1 \)