Sequences and series

Sequences

1. Write down the general term $u_n$ of the following sequence:
   

   (a)   $6,7,8,9,10,11,\dots$

   (b)   $5,10,15,20,25,30,\dots$

   (c)   $6,11,16,21,26,31,\dots$

   Solutions:    (a)  $u_n=n+5$;     (b)  $u_n=5n$;     (c)  $u_n=5n+1$
2. Write down the $n$th term of the following sequence:
   

   (a)   $1,4,7,10,13,\dots$

   (b)   $1,4,9,16,25,\dots$

   (c)   $1,4,16,64,256,\dots$

   Solutions:    (a)  $u_n=3n-2$;     (b)  $u_n=n^2$;     (c)  $u_n=4^{n-1}$
3. Write down the general term of the following sequence:
   

   (a)   $5,\frac{5}{2},\frac{5}{3},\frac{5}{4},1,\frac{5}{6},\dots$

   (b)   $\frac{1}{4},\frac{2}{5},\frac{3}{6},\frac{4}{7},\frac{5}{8},\dots$

   (c)   $-1,\frac{1}{2},-\frac{1}{3},\frac{1}{4},-\frac{1}{5},\frac{1}{6},\dots$

   Solutions:    (a)  $u_n=\frac{5}{n}$;     (b)  $u_n=\frac{n}{n+3}$;     (c)  $u_n=(-1{)}^n\cdot \frac{1}{n}$
4. Write down the recurrence relation between the terms of the following sequence:
   

   (a)   $1,4,7,10,13,\dots$

   (b)   $1,3,9,27,81,\dots$

   (c)   $\frac{1}{2},\frac{1}{4},\frac{1}{8},\frac{1}{16},\frac{1}{32},\dots$

   Solutions:    (a)  $u_{n+1}=u_n+3$;     (b)  $u_{n+1}=3u_n$;     (c)  $u_{n+1}=\frac{1}{2}{u}_n$
5. Write down the fourth and the seventh term of the following sequence:
   

   (a)   $u_n=n^2-5n+13$

   (b)   $u_{n+1}=2u_n,u_1=3$

   Solutions:    (a)  $u_4=9,u_7=27$;     (b)  $u_4=24,u_7=192$
6. Write down the first five terms of the following sequence and draw the graph:
   

   (a)   $u_n=\frac{3n}{n+1}$

   (b)   $u_n=\frac{4}{n^2+1}$

   Solutions:    (a)  $\frac{3}{2},2,\frac{9}{4},\frac{12}{5},\frac{5}{2},\dots$;     (b)  $2,\frac{4}{5},\frac{2}{5},\frac{4}{17},\frac{2}{13},\dots$
7. Write down the first five terms of the following sequence and draw the graph:
   

   (a)   $u_{n+1}=2u_n-3,u_1=4$

   (b)   $u_{n+1}=\frac{2}{3}{u}_n,u_1=27$

   Solutions:    (a)  $4,5,7,11,19,\dots$;     (b)  $27,18,12,8,\frac{16}{3},\dots$

Series and sigma notation

8. Find the sum of the first five terms of the following sequence:
   

   (a)   $u_n=4n-2$

   (b)   $u_n=(-1{)}^n\cdot n$

   Solutions:    (a)  $S_5=2+6+10+14+18=50$;     (b)  $S_5=-1+2-3+4-5=-3$
9. Find $S_4$ of the following sequence:
   

   (a)   $u_n=n^2$

   (b)   $u_{n+1}=\frac{1}{2}{u}_n,u_1=6$

   Solutions:    (a)  $S_4=1+4+9+16=30$;     (b)  $S_4=6+3+\frac{3}{2}+\frac{3}{4}=\frac{45}{4}$
10. Write down the terms and find the sum:
    

    (a)   ${\displaystyle \sum _{r=1}^3{r}^3}$

    (b)   ${\displaystyle \sum _{r=1}^4(r-1{)}^2}$

    Solutions:    (a)  $S_3=1+8+27=36$;     (b)  $S_4=0+1+4+9=14$
11. Find the sum:
    

    (a)   ${\displaystyle \sum _{r=1}^5(r^2-r)}$

    (b)   ${\displaystyle \sum _{r=6}^8(7r-13)}$

    Solutions:    (a)  $S_5=0+2+6+12+20=40$;     (b)  $29+36+43=108$
12. Write in sigma notation and calculate the sum:
    

    (a)   $7+13+19+25+31+37$

    (b)   $5+10+20+40+80+160+320$

    (c)   $\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\frac{4}{5}+\frac{5}{6}$

    Solutions:    (a)  ${\displaystyle \sum _{r=1}^6(6r+1)=132}$;     (b)  ${\displaystyle \sum _{r=1}^{7}5\cdot 2^{r-1}=635}$;     (c)  ${\displaystyle \sum _{r=1}^5}\frac{r}{r+1}=\frac{71}{20}$

Arithmetic sequence

13. ?
    ?

    An arithmetic sequence is a sequence of numbers in which the difference between two consecutive terms is constant.
    
    General term:  $u_n=u_1+(n-1)d$
    
    Common difference:  $d=u_2-u_1$
    Find the common difference, write down the general term and draw the graph of the following arithmetic sequence:
    

    (a)   $1,3,5,7,\dots$

    (b)   $5,2,-1,-4,\dots$

    (c)   $4,4,4,4,\dots$

    Solutions:    (a)  $u_n=2n-1$;     (b)  $u_n=-3n+8$;     (c)  $u_n=4$
14. Write down the general term of the following arithmetic sequence:
    

    (a)   $33,19,5,-9,\dots$

    (b)   $3\frac{1}{3},3\frac{2}{3},4,4\frac{1}{3},\dots$

    Solutions:    (a)  $u_n=-14n+47$;     (b)  $u_n=\frac{1}{3}n+3$
15. An arithmetic sequence has the first term $u_1=10$ and the common difference $d=4$. Write down the general term and the values of the first five terms.
    Solutions:    General term: $u_n=4n+6$,   the first five terms: $10,14,18,22,26$
16. An arithmetic sequence has the general term $u_n=17n-45$. Find the terms $u_6$ and $u_{13}$.
    Solutions:    $u_6=57,u_{13}=176$
17. An arithmetic sequence has the terms $100,87,74,61,\dots$   Find the terms $u_8$ and $u_{11}$.
    Solutions:    $u_8=9,u_{11}=-30$
18. An arithmetic sequence has terms $u_1=18$ and $u_7=66$. Write down the general term and find the value of the fifth term.
    Solutions:    $u_n=8n+10,u_5=50$
19. In an arithmetic sequence the third term is 38 and the tenth term is 10. Write down the general term.
    Solution:    $u_n=-4n+50$
20. In an arithmetic sequence the first term is 88 and the fourth term is 49. Write down the general term and find out how many terms are positive.
    Solutions:    $u_n=-13n+101$,   the first seven terms are positive
21. An arithmetic sequence has the terms $6,29,52,\dots$   Write down the general term and find out how many terms are less than 1000.
    Solutions:    $u_n=23n-17$,   the first 44 terms are less than 1000
22. Insert four numbers between the given numbers 20 and 50 so that you obtain a finite arithmetic sequence.
    Solutions:    AS:   $20,26,32,38,44,50$
23. Insert five numbers between the given numbers 5 and 21 so that you obtain a finite arithmetic sequence.
    Solutions:    AS:   $5,7\frac{2}{3},10\frac{1}{3},13,15\frac{2}{3},18\frac{1}{3},21$
24. ?
    ?

    Three numbers $a,b$ and $c$ form a finite arithmetic sequence if:
    
    $b=\frac{a+c}{2}$
    Find the value of $x$ for which the following three numbers form a finite arithmetic sequence:   $u_1=x-3,u_2=x+3,u_3=2x+1$
    Solutions:    $x=8$,    AS: 5, 11, 17
25. Find the value of $x$ for which the following three numbers form a finite arithmetic sequence:   $u_1=x+4,u_2=2x-4,u_3=\frac{x+1}{2}$
    Solutions:    $x=5$,    AS: 9, 6, 3
26. Find the value of $x$ so that the following numbers are the first three terms of an arithmetic sequence:   $u_1=x+2,u_2=3x+1,u_3=x^2$
    Solutions:    (1) $x=0$,    AS: 2, 1, 0;     (2) $x=5$,    AS: 7, 16, 25
27. Find the value of $x$ so that the following numbers are consecutive terms of an arithmetic sequence:   $u_1=\frac{1}{x},u_2=\frac{x}{2},u_3=\frac{x+1}{x}$
    Solutions:    (1) $x=2$,    AS: $\frac{1}{2},1,\frac{3}{2}$;     (2) $x=-1$,    AS: $-1,-\frac{1}{2},0$
28. Find the value of $x$ for which the following three numbers form a finite arithmetic sequence:   $u_1=2\cdot 5^x,u_2=5^{x+1},u_3=200$
    Solutions:    $x=2$,    AS: 50, 125, 200

Sum of the first n terms of an arithmetic sequence

29. ?
    ?

    The sum of the first $n$ terms of an arithmetic sequence can be calculated using the $s_n$ formula which has two forms:
    
    $s_n=\frac{n(u_1+u_n)}{2}$     or
    
    $s_n=\frac{n(2u_1+(n-1)d)}{2}$
    
    You can use $\sum$ notation on your calculator to calculate the sum, too.
    Find the sum of the first 10 terms of the arithmetic sequence:
    

    (a)   $u_n=6n$

    (b)   $u_n=3n+5$

    Solutions:    (a)  $S_{10}=330$;     (b)  $S_{10}=215$
30. Find the sum of the first 15 terms of the arithmetic sequence:
    

    (a)   $1,7,13,19,25,\dots$

    (b)   $123,112,101,90,79,\dots$

    (c)   $2,2\frac{1}{3},2\frac{2}{3},3,3\frac{1}{3},\dots$

    Solutions:    (a)  $S_{15}=645$;     (b)  $S_{15}=690$;     (c)  $S_{15}=65$
31. Calculate the sum:
    

    (a)   ${\displaystyle \sum _{r=1}^{20}(4r+9)}$

    (b)   ${\displaystyle \sum _{k=1}^{17}(499-9k)}$

    Solutions:    (a)  $S_{20}=1020$;     (b)  $S_{17}=7106$
32. An arithmetic sequence has the terms $u_1=17$ and $u_2=25$. Calculate the sum of the first 12 terms of this sequence.
    Solution:    $S_{12}=732$
33. An arithmetic sequence has the terms $u_1=17$ and $u_7=41$. Calculate the sum of the first 18 terms of this sequence.
    Solution:    $S_{18}=918$
34. In an arithmetic sequence the seventh term is 57 and the tenth term is 75. Find the common difference and the first term. Then calculate the sum of the first ten terms.
    Solutions:    $d=6,u_1=21,S_{10}=480$
35. Add up all the multiples of 7 which are less than 1000.
    Solution:    $S_{142}=71071$
36. Find the sum:   $103+113+123+133+\cdots +243$
    Solution:    $S_{15}=2595$
37. An arithmetic sequence has the general term $u_n=234-17n$. Calculate the sum of all positive terms of this sequence.
    Solution:    $S_{13}=1495$
38. An arithmetic sequence has the $n$th term $u_n=12n+34$. Calculate the sum of all terms which are less than 1000.
    Solution:    $S_{80}=41600$
39. The sum of the first 13 terms of an arithmetic sequence is 1144. The common difference of this sequence is 8. Find the first term of this sequence.
    Solution:    $u_1=40$
40. Twenty terms form a finite arithmetic sequence. The sum of this sequence is 3350 and the first term is 25. Find the last term of this sequence.
    Solution:    $u_{20}=310$
41. A finite arithmetic sequence has the first term 30 and the common difference 4. The sum of this sequence is 1054. Find the number of terms in this sequence.
    Solution:    $n=17$
42. Jamie decided to start the 30 Day Push Up Challenge. On the first day he must do 5 push ups. Each next day he must do 1 push up more than the previous day. Calculate the total amount of push ups Jamie has to do in 30 days.
    Solution:    $S_{30}=585$
43. On the 1st of September workers started digging a trench for an electric cable. On the first day they dug 300 meters of trench. On the second day they dug 320 meters, on the third day they dug 340 meters and so on: each next day they dug 20 meters more than the previous day. They finished the work on the 25th of September. Find the length of the trench they dug on the 25th of September and the total length of the trench they dug.
    Solutions:    $u_{25}=780\mathrm{m},S_{25}=13500\mathrm{m}$
44. Terms $u_1=4m+1$, $u_2=6m-3$ and $u_3=7m-2$ are the first three terms on an infinite arithmetic sequence.
    

    (a)   Find $m$.

    (b)   Write down values of $u_1,u_2$ and $u_3$ and find the common difference.

    (c)   Write the general term $u_n$.

    (d)   Find out which terms are less than 500.

    (e)   Calculate the sum of all terms which are less than 500.

    Solutions:    (a)  $m=5$;     (b)  $u_1=21,u_2=27,u_3=33;d=6$;     (c)  $u_n=6n+15$;     (d)  the first 80 terms;     (e)  $S_{80}=20640$

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