Sequences and series

Geometric sequence

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   A geometric sequence is a sequence of numbers in which the ratio between two consecutive terms is constant.
   
   General term:  $u_n=u_1{r}^{n-1}$
   
   Common ratio:  $r=\frac{u_2}{u_1}$
   Find the common ratio $r$ and write down the general term of the following geometric sequence:
   

   (a)   $3,6,12,24,\dots$

   (b)   $5,15,45,135,\dots$

   (c)   $80,40,20,10,\dots$

   Solutions:    (a)  $r=2,u_n=3\cdot 2^{n-1}$;     (b)  $r=3,u_n=5\cdot 3^{n-1}$;     (c)  $r=\frac{1}{2},u_n=80\cdot {\left(\frac{1}{2}\right)}^{n-1}$
2. Write down the general term and draw the graph of the following geometric sequence:
   

   (a)   $8,12,18,27,\dots$

   (b)   $8,-4,2,-1,\dots$

   Solutions:    (a)  $u_n=8\cdot {\left(\frac{3}{2}\right)}^{n-1}$;     (b)  $u_n=8\cdot {(-\frac{1}{2})}^{n-1}$
3. Write down the general term of the following geometric sequence:
   

   (a)   $-2,2,-2,2,\dots$

   (b)   $\frac{9}{2},\frac{3}{4},\frac{1}{8},\frac{1}{48},\dots$

   (c)   $\sqrt{2},2,2\sqrt{2},4,\dots$

   Solutions:    (a)  $u_n=-2\cdot (-1{)}^{n-1}=2\cdot (-1{)}^n$;     (b)  $u_n=\frac{9}{2}\cdot {\left(\frac{1}{6}\right)}^{n-1}$;     (c)  $u_n=\sqrt{2}\cdot (\sqrt{2}{)}^{n-1}={\sqrt{2}}^n$
4. A geometric sequence has the first term $u_1=-3$ and the common ratio $r=4$. Write down the general term and the first five terms.
   Solutions:    General term: $u_n=-3\cdot 4^{n-1}$;    the first five terms:   $-3,-12,-48,-192,-768$
5. A geometric sequence has the general term $u_n=6\cdot 5^{n-1}$. Find the terms $u_4$ and $u_7$.
   Solutions:    $u_4=750,u_7=93750$
6. A geometric sequence has the terms $5120,2560,1280,640\dots$   Find the terms $u_7,u_{11}$ and $u_{15}$.
   Solutions:    $u_7=80,u_{11}=5,u_{15}=\frac{5}{16}$
7. A geometric sequence has the terms $u_1=7$ and $u_4=189$. Find the terms $u_5$ and $u_6$.
   Solutions:    $u_5=567,u_6=1701$
8. In a geometric sequence the third term is 36 and the sixth term is 288. Find the first term and the common ratio.
   Solutions:    $u_1=9,r=2$
9. The second term in a geometric sequence is 3 and the seventeenth term is $147\sqrt{7}$. Find the value of $u_{20}$.
   Solution:    $u_{20}=1029$
10. A geometric sequence has the first term $u_1=4$ and the common ratio $r=3$. Find out how many terms are less than 1000.
    Solution:    The first 6 terms are less than 1000.
11. A geometric sequence has the terms $\sqrt{3},3,3\sqrt{3},9,\dots$   Write down the general term and find out how many terms are less than $10000$.
    Solutions:    $u_n={\sqrt{3}}^n$,    the first 16 terms are less than $10000$.
12. A geometric sequence has the terms $u_1=2\sqrt{2}$ and $u_2=2$. Write down the general term and find out how many terms are greater than ${10}^{-6}$.
    Solutions:    $u_n=4\cdot {\left(\frac{\sqrt{2}}{2}\right)}^n$,    the first 43 terms are greater than ${10}^{-6}$.
13. Insert four numbers between the given numbers 15 and 480 so that you obtain a finite geometric sequence.
    Solutions:    GS: $15,30,60,120,240,480$
14. Insert three positive numbers between the given numbers 256 and 625 so that you obtain a finite geometric sequence.
    Solutions:    GS: $256,320,400,500,625$
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    Three numbers $a,b$ and $c$ form a finite geometric sequence if:
    
    $b^2=ac$
    Find the value of $x$ for which the following three numbers form a finite geometric sequence:   $u_1=x-5,u_2=x-1,u_3=2x+4$
    Solutions:    (1) $x_1=7$,   GS: $2,6,18$;     (2) $x_2=-3$,   GS: $-8,-4,-2$
16. Find the value of $x$ for which the following three numbers form a finite geometric sequence:   $u_1=x-7,u_2=x+3,u_3=4x-3$
    Solutions:    (1) $x_1=12$,   GS: $5,15,45$;     (2) $x_2=\frac{1}{3}$,   GS: $-\frac{20}{3},\frac{10}{3},-\frac{5}{3}$
17. Find the value of $x$ so that the following numbers are the first three terms of a geometric sequence:   $u_1=\frac{x-7}{9},u_2=\frac{x-4}{6},u_3=\frac{x}{4}$
    Solutions:    $x_1=16$,   GS: $1,2,4$
18. Find the value of $x$ so that the following numbers are consecutive terms of a geometric sequence:   $u_1=2,u_2=2+\sqrt{x},u_3=x-158$
    Solutions:    $x_1=400$,   GS: $2,22,242$

Sum of the first n terms of a geometric sequence

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    The sum of the first $n$ terms of a geometric sequence can be calculated using the $s_n$ formula which has two forms:
    
    $s_n=u_1\frac{1-r^n}{1-r}$     or
    
    $s_n=u_1\frac{r^n-1}{r-1}$
    
    You can use $\sum$ notation on your calculator to calculate the sum, too.
    Find the sum of the first 6 terms of the geometric sequence:
    

    (a)   $u_n=8\cdot 3^{n-1}$

    (b)   $u_n=5^n$

    Solutions:    (a)  $S_6=2912$;     (b)  $S_6=19 530$
20. Find the sum of the first 7 terms of the geometric sequence:
    

    (a)   $5,15,45,135,\dots$

    (b)   $1280,640,320,160,\dots$

    (c)   $3645,-2430,1620,-1080,\dots$

    Solutions:    (a)  $S_7=5465$;     (b)  $S_7=2540$;     (c)  $S_7=2315$
21. Find the sum of the first 8 terms of the geometric sequence and write the result in exact form:
    

    (a)   $6,2,\frac{2}{3},\frac{2}{9},\dots$

    (b)   $2,2\sqrt{2},4,4\sqrt{2},\dots$

    Solutions:    (a)  $S_8=\frac{6560}{729}$;     (b)  $S_8=30+30\sqrt{2}$
22. Calculate the sum:
    

    (a)   ${\displaystyle \sum _{r=1}^{15}{2}^r}$

    (b)   ${\displaystyle \sum _{r=1}^8{\left(\frac{3}{2}\right)}^{r-1}}$

    Solutions:    (a)  $S_{15}=65 534$;     (b)  $S_8=\frac{6305}{128}$
23. In a geometric sequence the first term is 3 and the seventh term is 15. Find the sum of the first 20 terms. Round the result to five significant digits.
    Solution:    $S_{20}=2074.5$
24. A geometric sequence has the general term $u_n=5\cdot 3^{n-1}$. Find the sum of all terms which are less than 100 000.
    Solution:    $S_{10}=147620$
25. A geometric sequence has the terms $u_1=25$ and $u_2=30$. Find the sum of all terms which are less than 100. Round your result to 4 significant figures.
    Solution:    $S_8=412.5$
26. The sum of the first 6 terms of a geometric sequence is 9555. The common ratio of this sequence is 4. Find the first term of this sequence.
    Solution:    $u_1=7$
27. A finite geometric sequence has the first term 4 and the second term 20. The sum of this sequence is 78 124. Find the number of terms in this sequence.
    Solution:    $n=7$

Compound interest

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    Adding interest to your bank deposit can be modelled using the geometric sequence. If the interest is compounded yearly, the value of your deposit after $n$ years will be:
    
    $u_n=u_0{(1+\frac{p}{100})}^n$     or
    
    $FV=PV{(1+\frac{p}{100})}^n$
    
    If your calculator has the Finance Solver function, you can use it for all calculations in this section.
    John Smith deposits 1000 € in his account at ABC-Bank. This bank compounds his deposit annually at 7% interest rate per annum (p.a.). How much will he have in his account:
    

    (a)   after 1 year,

    (b)   after 2 years,

    (c)   after 5 years?

    Solutions:    (a)  1070 €;     (b)  1144,90 €;     (c)  1402,55 €
29. Jane Doe deposits 14 790 € in The Cayman National Bank. The interest rate is 9% p.a. compounding annually. Find the final amount of her deposit after 8 years. How much will be the total interest?
    Solutions:    She will have 29 470 €. The total interest will be 14 680 €.
30. Joe Bloggs deposited 3292 £ in his savings account. After 10 years he found out that he has 4873 £ in his account. Calculate the annual interest rate.
    Solution:    The interest rate is 4% (compounded annually).
31. Uncle Sam deposited 15 800 $ in his bank account. The annual interest rate is 6% (compounded annually). In how many years will he have 33 700 $ in his account?
    Solution:    In 13 years.
32. John Bull invested 52 500 £ in stock with annual growth rate of 7.5%. In how many years will he have the amount of 87 100 £?
    Solution:    In 7 years.
33. The Best National Bank compounds deposits annually at 8% (p.a.). How many years would it take for your deposit to double? Round the number of years to nearest integer.
    Solution:    It'll take 9 years.
34. A bank compounds deposits annually at 7% (p.a.). How many years would it take for your deposit to increase by 50%? Round the number of years to nearest integer.
    Solution:    It'll take 6 years.
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    If the interest is compounded $k$ times in a year, the value of your deposit after $n$ years will be:
    
    $FV=PV{(1+\frac{p}{100k})}^{kn}$
    
    Finance Solver can solve this, too.
    You deposited 12 000 € in your bank account. The interest rate is 6% (p.a.). How much will you have in your account after five years:
    

    (a)   if the interest is compounded yearly,

    (b)   if the interest is compounded monthly?

    Solutions:    (a)  16 058.71 €;     (b)  16 186.20 €
36. Ivan Ivanovich Ivanov deposited 20 000 € in his account at The Swiss Royal Bank. The interest rate is 5% (p.a.). How much will he have in his account after ten years:
    

    (a)   if the interest is compounded yearly,

    (b)   if the interest is compounded half-yearly,

    (c)   if the interest is compounded quarterly?

    Solutions:    (a)  32 577.89 €;     (b)  32 772.33 €;     (c)  32 872.39 €

37. Our friend wants to buy a car but she doesn't have enough money, so she is going to take out a loan of 15 000 €. The bank charges interest at a rate of 5% p.a. Calculate her monthly payment if she plans to repay the loan:
    

    (a)   in three years,

    (b)   in six yeaars.

    Solutions:    (a)  449.56 €;     (b)  241.57 €
38. Annie wants to buy a small apartment. She is going to take out a loan of 50 000 € and she plans to repay it in 10 years. Her bank charges interest at a rate of 6% p.a.
    

    (a)   Calculate her monthly payment.

    (b)   Calculate the total amount of all her payments.

    Benny wants to buy a small apartment, too. He is going to take out a loan of 50 000 € and he plans to repay it in 15 years. His bank charges interest at a rate of 5% p.a.

    (c)   Calculate his monthly payment.

    (d)   Calculate the total amount of all his payments.

    Solutions:    (a)  555.10 €;     (b)  66 612.30 €;     (c)  395.40 €;     (d)  71 171.43 €
39. Anastasia wanted to save some money. She decided to deposit 95 € in the beginning of each month for the next 5 years. Her bank has the interest rate of 2% p.a.
    

    (a)   Calculate the total amount she has in her bank after 5 years.

    (b)   Calculate the total amount of her deposits.

    (c)   Calculate the total amount of interest she will get.

    Boris decided to take out a loan of 6 000 €. He will repay it in 5 years in monthly payments. His bank charges interest at a rate of 2% p.a.

    (d)   Calculate his monthly payment.

    (e)   Calculate the total amount he will repay to his bank.

    (f)   Calculate the amount of interest he will have to pay.

    Solutions:    (a)  5 999.48 €;     (b)  5 700 €;     (c)  299.48 €;     (d)  105.17 €;     (e)  6 309.99 €;     (f)  309.99 €

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