Domov

Sequences and series

Geometric sequences

  1. Find the common ratio \(r\) and write down the general term of the following geometric sequence:

    (a)   \(3,~ 6,~ 12,~ 24,~\ldots\)

    (b)   \(5,~ 15,~ 45,~ 135,~\ldots\)

    (c)   \(80,~ 40,~ 20,~ 10,~\ldots\)

    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)   \(8,~ 12,~ 18,~ 27,~\ldots\)

    (b)   \(8,~ -4,~ 2,~ -1,~\ldots\)

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

    (a)   \(-2,~ 2,~ -2,~ 2,~\ldots\)

    (b)   \(\frac{9}{2},~ \frac{3}{4},~ \frac{1}{8},~ \frac{1}{48},~\ldots\)

    (c)   \(\sqrt{2},~~ 2,~~ 2\sqrt{2},~~ 4,~\ldots\)

    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)  \(u_n=\sqrt{2}\cdot\big(\sqrt{2}\big)^{n-1}=\sqrt{2}^{~n}\)
  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: \(u_n=-3\cdot4^{n-1}\);    the first five terms:   \(-3,~ -12,~ -48,~ -192,~ -768\)
  5. A geometric sequence has the general term \(u_n=6\cdot5^{n-1}\). Find the terms \(u_4\) and \(u_7\).
    Solutions:    \(u_4=750,~~ u_7=93\,750\)
  6. A geometric sequence has the terms \(5120,~ 2560,~ 1280,~ 640~\ldots\). 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 \(u_{20}\).
    Solutions:    \(u_{20}=1029\)
  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.
    Solutions:    The first 6 terms are less than 1000.
  11. A geometric sequence has the terms \(\sqrt{3},~3,~ 3\sqrt{3},~9,~\ldots\)   Write down the general term and find out how many terms are less than \(10\,000\).
    Solutions:    \(u_n=\sqrt{3}^{~n}\),    the first 16 terms are less than \(10\,000\).
  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: \(256,~ 320,~ 400,~ 500,~ 625\)
  15. 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:   \(u_1=\frac{\textstyle x-7}{\textstyle 9},~~ u_2=\frac{\textstyle x-4}{\textstyle 6},~~ u_3=\frac{\textstyle x}{\textstyle 4}\)
    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: \(2,~ 22,~ 242\)

Compound interest

  1. 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 €
  2. 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 €.
  3. 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.
    Solutions:    The interest rate is 4% (compounded annually).
  4. 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?
    Solutions:    In 13 years.
  5. 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 £?
    Solutions:    In 7 years.
  6. 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.
    Solutions:    It'll take 9 years.
  7. 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.
    Solutions:    It'll take 6 years.

Sum of the first n terms of a geometric sequence

  1. Find the sum of the first 6 terms of the geometric sequence:

    (a)   \(u_n=8\cdot3^{n-1}\)

    (b)   \(u_n=5^n\)

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

    (a)   \(5,~ 15,~ 45,~ 135,~\ldots\)

    (b)   \(1280,~ 640,~ 320,~ 160,~\ldots \)

    (c)   \(3645,~ -2430,~ 1620,~ -1080,~\ldots \)

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

    (a)   \(6,~ 2,~ \frac{2}{3},~ \frac{2}{9},~\ldots\)

    (b)   \(2,~ 2\sqrt{2},~ 4,~ 4\sqrt{2},~\ldots\)

    Solutions:    (a)  \(S_8=\frac{6560}{729}\);     (b)  \(S_8=30+30\sqrt{2}\)
  4. 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)  \(S_8=\frac{6305}{128}\)
  5. 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.
    Solutions:    \(S_{20}=2074,\!5\)
  6. 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.
    Solutions:    \(S_{10}=147 620\)
  7. A geometric sequence has the terms \(u_1=25\) and \(u_2=30\). Find the sum of all terms which are less than 100.
    Solutions:    \(S_8=412,\!47712\)
  8. 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.
    Solutions:    \(u_1=7\)
  9. 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.
    Solutions:    \(n=7\)

Sum to infinity of a geometric sequence

  1. Find the sum to infinity of the following geometric sequences:

    (a)   \(1,~ \frac{1}{3},~ \frac{1}{9},~ \frac{1}{27},~\ldots\)

    (b)   \(32,~ -8,~ 2,~ -\frac{1}{2},~\ldots\)

    (c)   \(24,~ 36,~ 54,~ 81,~\ldots\)

    Solutions:    (a)  \(S_\infty=\frac{3}{2}\);     (b)  \(S_\infty=25\frac{3}{5}\);     (c)  The sum doesn't exist \((S_\infty=\infty)\).
  2. Calculate the sum:

    (a)   \({\displaystyle\sum_{n=1}^\infty \left(\frac{1}{4}\right)^n}\)

    (b)   \({\displaystyle\sum_{n=0}^\infty \left(\frac{3}{5}\right)^n}\)

    Solutions:    (a)  \(S=\frac{1}{3}\);     (b)  \(S=\frac{5}{2}\)
  3. A geometric sequence has the common ratio \(\frac{1}{5}\) and the sum to infinity 10. Find the first term of this sequence.
    Solutions:    \(u_1=8\)
  4. A geometric sequence has the first term 6 and the sum to infinity 18. Find the common ratio of this sequence.
    Solutions:    \(r=\frac{2}{3}\)
  5. Express the following recurring decimal as a simplified fraction:

    (a)   \(0,\!\dot{7}\)

    (b)   \(0,\!8\dot{3}\)

    (c)   \(0,\!\dot{4}\dot{2}\)

    Solutions:    (a)  \(\frac{7}{9}\);     (b)  \(\frac{5}{6}\);     (c)  \(\frac{14}{33}\)

Factorial and binomial coefficient

  1. Calculate:

    (a)   \(5!\)

    (b)   \(4!\)

    (c)   \(3!\)

    (d)   \(2!\)

    (e)   \(1!\)

    (f)   \(0!\)

    Solutions:    (a)  \(120\);     (b)  \(24\);     (c)  \(6\);     (d)  \(2\);     (e)  \(1\);     (f)  \(1\)
  2. A photographer wants to take a photo of three girls: Alice, Betty and Cindy. He's going to arrange them in a line. In how many ways can he line up the girls?
    Solutions:    \(3!=6\).   In 6 ways.
  3. Consider arranging letters A, B, C, D. In how many ways can this be done?
    Solutions:    In 24 ways.
  4. Today Judy has five things to do: she must go to the bank, to the dentist, to the library, to the museum and to the shopping centre. In how many different ways can she select the order of her appointments?
    Solutions:    In 120 ways.
  5. Jamie's favourite books are: Alchemist, Beowulf, Catch 22 and Dracula. He wants to take two of his favourite books on a voyage. In how many different ways can he select them?
    Solutions:    \({4\choose 2}=6\).   In 6 ways.
  6. Little Annie knows there are 10 different dolls in the toy shop. She would like to buy them all, but she has enough money to buy only two of them. In how many ways can she pick them?
    Solutions:    In 45 ways.
  7. There are 20 students in a class. They must select 3 representatives for an upcoming meeting. In how many ways can the representatives be chosen?
    Solutions:    In 1140 ways.
  8. Evaluate the following binomial coefficients:

    (a)   \({\displaystyle {6\choose3}}\)

    (b)   \({\displaystyle {10\choose7}}\)

    (c)   \({\displaystyle {33\choose5}}\)

    Solutions:    (a)  \(20\);     (b)  \(120\);     (c)  \(237\,336\)
  9. Calculate and compare the values:

    (a)   \({\displaystyle {8\choose1}}\)   and   \({\displaystyle {8\choose7}}\)

    (b)   \({\displaystyle {8\choose2}}\)   and   \({\displaystyle {8\choose6}}\)

    (c)   \({\displaystyle {8\choose0}}\)   and   \({\displaystyle {8\choose8}}\)

    Solutions:    (a)  \(8\);     (b)  \(28\);     (c)  \(1\)
  10. Given the set \(\mathcal{A}=\{1,2,3,4\}\), find all the subsets of \(\mathcal{A}\) which

    (a)   contain 0 elements,

    (b)   contain 1 element,

    (c)   contain 2 elements,

    (d)   contain 3 elements,

    (e)   contain 4 elements.

    Solutions:    (a)  \(\{\}\);     (b)  \(\{1\},\{2\},\{3\},\{4\}\);     (c)  \(\{1,2\},\{1,3\},\{1,4\},\{2,3\},\{2,4\},\{3,4\}\);     (d)  \(\{1,2,3\},\{1,2,4\},\{1,3,4\},\{2,3,4\}\);     (e)  \(\{1,2,3,4\}\)

Binomial expansions

  1. Expand the following expressions:

    (a)   \((x+1)^3\)

    (b)   \((x+1)^4\)

    Solutions:    (a)  \(x^3+3x^2+3x+1\);     (b)  \(x^4+4x^3+6x^2+4x+1\)
  2. Expand the following expressions:

    (a)   \((x+5)^3\)

    (b)   \((2x-3)^4\)

    (c)   \((3x-4)^5\)

    Solutions:    (a)  \(x^3+15x^2+75x+125\);     (b)  \(16x^4-96x^3+216x^2-216x+81\);     (c)  \(243x^5-1620x^4+4320x^3-5760x^2+3840x-1024\)
  3. Simplify the following expressions (write the result in exact form):

    (a)   \((1+\sqrt{2})^4\)

    (b)   \((2-\sqrt{3})^5\)

    Solutions:    (a)  \(17+12\sqrt{2}\);     (b)  \(362-209\sqrt{3}\)
  4. Find the term containing \(x^4\) in following expansions:

    (a)   \((7+x)^5\)

    (b)   \((2+3x)^6\)

    Solutions:    (a)  \(35x^4\);     (b)  \(4860x^4\)
  5. Find the coefficient of the term containing \(x^5\) in following expansions:

    (a)   \((-2+x)^{12}\)

    (b)   \((3x+\sqrt{5})^8\)

    Solutions:    (a)  \(-101\,376\);     (b)  \(68\,040\,\sqrt{5}\)
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