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Sura ya 12 Mazoezi Mapitio

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    Utaratibu

    Zoezi\(\PageIndex{1}\) Write the First Few Terms of a Sequence

    Katika mazoezi yafuatayo, weka maneno matano ya kwanza ya mlolongo ambao muda wake umetolewa.

    1. \(a_{n}=7 n-5\)
    2. \(a_{n}=3^{n}+4\)
    3. \(a_{n}=2^{n}+n\)
    4. \(a_{n}=\frac{2 n+1}{4^{n}}\)
    5. \(a_{n}=\frac{(-1)^{n}}{n^{2}}\)
    Jibu

    2. \(7,13,31,85,247\)

    4. \(\frac{3}{4}, \frac{5}{16}, \frac{7}{64}, \frac{9}{256}, \frac{11}{1024}\)

    Zoezi\(\PageIndex{2}\) Find a Formula for the General Term (\(n\)th Term of a Sequence

    Katika mazoezi yafuatayo, pata muda wa jumla kwa mlolongo ambao maneno matano ya kwanza yanaonyeshwa.

    1. \(9,18,27,36,45, \dots\)
    2. \(-5,-4,-3,-2,-1, \dots\)
    3. \(\frac{1}{e^{3}}, \frac{1}{e^{2}}, \frac{1}{e}, 1, e, \ldots\)
    4. \(1,-8,27,-64,125, \ldots\)
    5. \(-\frac{1}{3},-\frac{1}{2},-\frac{3}{5},-\frac{2}{3},-\frac{5}{7}, \dots\)
    Jibu

    1. \(a_{n}=9 n\)

    3. \(a_{n}=e^{n-4}\)

    5. \(a_{n}=-\frac{n}{n+2}\)

    Zoezi\(\PageIndex{3}\) Use Factorial Notation

    Katika mazoezi yafuatayo, kwa kutumia notation factorial, kuandika maneno tano ya kwanza ya mlolongo ambao muda wake mkuu hutolewa.

    1. \(a_{n}=4 n !\)
    2. \(a_{n}=\frac{n !}{(n+2) !}\)
    3. \(a_{n}=\frac{(n-1) !}{(n+1)^{2}}\)
    Jibu

    2. \(\frac{1}{6}, \frac{1}{12}, \frac{1}{20}, \frac{1}{30}, \frac{1}{42}\)

    Zoezi\(\PageIndex{4}\) Find the Partial Sum

    Katika mazoezi yafuatayo, panua jumla ya sehemu na kupata thamani yake.

    1. \(\sum_{i=1}^{7}(2 i-5)\)
    2. \(\sum_{i=1}^{3} 5^{i}\)
    3. \(\sum_{k=0}^{4} \frac{4}{k !}\)
    4. \(\sum_{k=1}^{4}(k+1)(2 k+1)\)
    Jibu

    1. \(\begin{array}{l}{-3+(-1)+1+3+5} {+7+9=21}\end{array}\)

    3. \(4+4+2+\frac{2}{3}+\frac{1}{6}=\frac{65}{6}\)

    Zoezi\(\PageIndex{5}\) Use Summation Notation to Write a Sum

    Katika mazoezi yafuatayo, weka kila jumla kwa kutumia maelezo ya muhtasari.

    1. \(-\frac{1}{3}+\frac{1}{9}-\frac{1}{27}+\frac{1}{81}-\frac{1}{243}\)
    2. \(4-8+12-16+20-24\)
    3. \(4+2+\frac{4}{3}+1+\frac{4}{5}\)
    Jibu

    1. \(\sum_{n=1}^{5}(-1)^{n} \frac{1}{3^{n}}\)

    3. \(\sum_{n=1}^{5} \frac{4}{n}\)

    Utaratibu wa hesabu

    Zoezi\(\PageIndex{6}\) Determine if a Sequence is Arithmetic

    Katika mazoezi yafuatayo, onyesha kama kila mlolongo ni hesabu, na ikiwa ni hivyo, onyesha tofauti ya kawaida.

    1. \(1,2,4,8,16,32, \dots\)
    2. \(-7,-1,5,11,17,23, \dots\)
    3. \(13,9,5,1,-3,-7, \dots\)
    Jibu

    2. Mlolongo ni hesabu na tofauti ya kawaida\(d=6\).

    Zoezi\(\PageIndex{7}\) Determine if a Sequence is Arithmetic

    Katika mazoezi yafuatayo, weka maneno matano ya kwanza ya kila mlolongo wa hesabu na neno la kwanza na tofauti ya kawaida.

    1. \(a_{1}=5\)na\(d=3\)
    2. \(a_{1}=8\)na\(d=-2\)
    3. \(a_{1}=-13\)na\(d=6\)
    Jibu

    1. \(5,8,11,14,17\)

    3. \(-13,-7,-1,5,11\)

    Zoezi\(\PageIndex{8}\) Find the General Term (\(n\)th Term) of an Arithmetic Sequence

    Katika mazoezi yafuatayo, tafuta neno lililoelezwa kwa kutumia habari iliyotolewa.

    1. Pata muda wa ishirini na tano wa mlolongo ambapo muda wa kwanza ni tano na tofauti ya kawaida ni tatu.
    2. Kupata muda thelathini ya mlolongo ambapo muda wa kwanza ni\(16\) na tofauti ya kawaida ni\(−5\).
    3. Kupata kumi na saba mrefu ya mlolongo ambapo muda wa kwanza ni\(−21\) na tofauti ya kawaida ni mbili.
    Jibu

    2. \(-129\)

    Zoezi\(\PageIndex{9}\) Find the General Term (\(n\)th Term) of an Arithmetic Sequence

    Katika mazoezi yafuatayo, tafuta neno lililoonyeshwa na upe fomu kwa muda mrefu.

    1. Kupata kumi na nane mrefu ya mlolongo ambapo mrefu tano ni\(12\) na tofauti ya kawaida ni saba.
    2. Pata muda wa ishirini na moja ya mlolongo ambapo muda wa saba ni\(14\) na tofauti ya kawaida ni\(−3\).
    Jibu

    1. \(a_{18}=103 .\)Neno la jumla ni\(a_{n}=7 n-23\).

    Zoezi\(\PageIndex{10}\) Find the General Term (\(n\)th Term) of an Arithmetic Sequence

    Katika mazoezi yafuatayo, tafuta neno la kwanza na tofauti ya kawaida ya mlolongo na masharti yaliyotolewa. Kutoa formula kwa muda wa jumla.

    1. Muda wa tano ni\(17\) na muda wa kumi na nne ni\(53\).
    2. Muda wa tatu ni\(−26\) na muda wa kumi na sita ni\(−91\).
    Jibu

    1. \(a_{1}=1, d=4 .\)Neno la jumla ni\(a_{n}=4 n-3\).

    Zoezi\(\PageIndex{11}\) Find the Sum of the First \(n\) Terms of an Arithmetic Sequence

    Katika mazoezi yafuatayo, pata jumla ya\(30\) maneno ya kwanza ya kila mlolongo wa hesabu.

    1. \(7,4,1,-2,-5, \dots\)
    2. \(1,6,11,16,21, \ldots\)
    Jibu

    1. \(-430\)

    Zoezi\(\PageIndex{12}\) Find the Sum of the First \(n\) Terms of an Arithmetic Sequence

    Katika mazoezi yafuatayo, tafuta jumla ya masharti kumi na tano ya mlolongo wa hesabu ambao muda wake umetolewa.

    1. \(a_{n}=4 n+7\)
    2. \(a_{n}=-2 n+19\)
    Jibu

    1. \(585\)

    Zoezi\(\PageIndex{13}\) Find the Sum of the First \(n\) Terms of an Arithmetic Sequence

    Katika mazoezi yafuatayo, tafuta kila jumla.

    1. \(\sum_{i=1}^{50}(4 i-5)\)
    2. \(\sum_{i=1}^{30}(-3 i-7)\)
    3. \(\sum_{i=1}^{35}(i+10)\)
    Jibu

    1. \(4850\)

    3. \(980\)

    Utaratibu wa kijiometri na mfululizo

    Zoezi\(\PageIndex{14}\) Determine if a Sequence is Geometric

    Katika mazoezi yafuatayo, onyesha kama mlolongo ni kijiometri, na ikiwa ni hivyo, onyesha uwiano wa kawaida.

    1. \(3,12,48,192,768,3072, \dots\)
    2. \(5,10,15,20,25,30, \dots\)
    3. \(112,56,28,14,7, \frac{7}{2}, \ldots\)
    4. \(9,-18,36,-72,144,-288, \dots\)
    Jibu

    2. Mlolongo sio kijiometri.

    4. Mlolongo ni kijiometri na uwiano wa kawaida\(r=−2\).

    Zoezi\(\PageIndex{15}\) Determine if a Sequence is Geometric

    Katika mazoezi yafuatayo, weka maneno tano ya kwanza ya kila mlolongo wa kijiometri na uwiano wa kwanza uliopewa na uwiano wa kawaida.

    1. \(a_{1}=-3\)na\(r=5\)
    2. \(a_{1}=128\)na\(r=\frac{1}{4}\)
    3. \(a_{1}=5\)na\(r=-3\)
    Jibu

    2. \(128,32,8,2, \frac{1}{2}\)

    Zoezi\(\PageIndex{16}\) Find the General Term (\(n\)th Term) of a Geometric Sequence

    Katika mazoezi yafuatayo, tafuta muda ulioonyeshwa wa mlolongo ambapo muda wa kwanza na uwiano wa kawaida hutolewa.

    1. Kupata\(a_{9}\) aliyopewa\(a_{1}=6\) na\(r=2\)
    2. Kupata\(a_{11}\) aliyopewa\(a_{1}=10,000,000\) na\(r=0.1\)
    Jibu

    1. \(1,536\)

    Zoezi\(\PageIndex{17}\) Find the General Term (\(n\)th Term) of a Geometric Sequence

    Katika mazoezi yafuatayo, tafuta muda ulioonyeshwa wa mlolongo uliopewa. Pata muda wa jumla wa mlolongo.

    1. Kupata\(a_{12}\) ya mlolongo,\(6,-24,96,-384,1536,-6144, \dots\)
    2. Kupata\(a_{9}\) ya mlolongo,\(4374,1458,486,162,54,18, \ldots\)
    Jibu

    1. \(a_{12}=-25,165,824 .\)Neno la jumla ni\(a_{n}=6(-4)^{n-1}\)

    Zoezi\(\PageIndex{18}\) Find the Sum of the First \(n\) terms of a Geometric Sequence

    Katika mazoezi yafuatayo, pata jumla ya maneno kumi na tano ya kila mlolongo wa kijiometri.

    1. \(-4,8,-16,32,-64,128 \ldots\)
    2. \(3,12,48,192,768,3072 \ldots\)
    3. \(3125,625,125,25,5,1 \ldots\)
    Jibu

    1. \(5,460\)

    3. \(\approx 3906.25\)

    Zoezi\(\PageIndex{19}\) find the Sum of the First \(n\) terms of a Geometric Sequence

    Katika mazoezi yafuatayo, pata jumla

    1. \(\sum_{i=1}^{8} 7(3)^{i}\)
    2. \(\sum_{i=1}^{6} 24\left(\frac{1}{2}\right)^{i}\)
    Jibu

    2. \(\frac{189}{8}=23.625\)

    Zoezi\(\PageIndex{20}\) Find the Sum of an Infinite Geometric Series

    Katika mazoezi yafuatayo, pata jumla ya kila mfululizo wa kijiometri usio na kipimo.

    1. \(1-\frac{1}{3}+\frac{1}{9}-\frac{1}{27}+\frac{1}{81}-\frac{1}{243}+\frac{1}{729}-\dots\)
    2. \(49+7+1+\frac{1}{7}+\frac{1}{49}+\frac{1}{343}+\ldots\)
    Jibu

    2. \(\frac{343}{6} \approx 57.167\)

    Zoezi\(\PageIndex{21}\) Find the Sum of an Infinite Geometric Series

    Katika mazoezi yafuatayo, andika kila decimal kurudia kama sehemu.

    1. \(0 . \overline{8}\)
    2. \(0 . \overline{36}\)
    Jibu

    2. \(\frac{4}{11}\)

    Zoezi\(\PageIndex{22}\) Apply Geometric Sequences and Series in the Real World

    Katika mazoezi yafuatayo, tatua tatizo.

    1. Je, ni athari ya jumla juu ya uchumi wa marupurupu ya kodi ya serikali ya $\(360\) kwa kila kaya ili kuchochea uchumi kama kila kaya itatumia\(60\)% ya marupurupu katika bidhaa na huduma?
    2. Adam alipata kazi yake ya kwanza ya wakati wote baada ya kuhitimu shule ya sekondari akiwa na umri wa miaka 17. Aliamua kuwekeza $\(300\) kwa mwezi katika IRA (annuity). Maslahi juu ya annuity\(7\) ni% ambayo imezungukwa kila mwezi. Ni kiasi gani katika akaunti ya Adamu atakapostaafu katika siku yake ya kuzaliwa sitini na saba?
    Jibu

    2. \(\$ 1,634,421.27\)

    Theorem ya Binomial

    Zoezi\(\PageIndex{23}\) Use Pascal's Triangle to Expand a Binomial

    Katika mazoezi yafuatayo, panua kila binomial kwa kutumia Triangle ya Pascal.

    1. \((a+b)^{7}\)
    2. \((x-y)^{4}\)
    3. \((x+6)^{3}\)
    4. \((2 y-3)^{5}\)
    5. \((7 x+2 y)^{3}\)
    Jibu

    2. \(x^{4}-4 x^{3} y+6 x^{2} y^{2}-4 x y^{3}+y^{4}\)

    4. \(\begin{array}{l}{32 y^{5}-240 y^{4}+720 y^{3}-1080 y^{2}} {+810 y-243}\end{array}\)

    Zoezi\(\PageIndex{24}\) Evaluate a Binomial Coefficient

    Katika mazoezi yafuatayo, tathmini.

      1. \(\left( \begin{array}{l}{11} \\ {1}\end{array}\right)\)
      2. \(\left( \begin{array}{l}{12} \\ {12}\end{array}\right)\)
      3. \(\left( \begin{array}{l}{13} \\ {0}\end{array}\right)\)
      4. \(\left( \begin{array}{l}{8} \\ {3}\end{array}\right)\)
      1. \(\left( \begin{array}{l}{7} \\ {1}\end{array}\right)\)
      2. \(\left( \begin{array}{l}{5} \\ {5}\end{array}\right)\)
      3. \(\left( \begin{array}{l}{9} \\ {0}\end{array}\right)\)
      4. \(\left( \begin{array}{l}{9} \\ {5}\end{array}\right)\)
      1. \(\left( \begin{array}{l}{1} \\ {1}\end{array}\right)\)
      2. \(\left( \begin{array}{l}{15} \\ {15}\end{array}\right)\)
      3. \(\left( \begin{array}{l}{4} \\ {0}\end{array}\right)\)
      4. \(\left( \begin{array}{l}{11} \\ {2}\end{array}\right)\)
    Jibu

    1.

    1. \(11\)
    2. \(1\)
    3. \(1\)
    4. \(56\)

    3.

    1. \(1\)
    2. \(1\)
    3. \(1\)
    4. \(55\)
    Zoezi\(\PageIndex{25}\) Use the Binomial Theorem to Expand a Binomial

    Katika mazoezi yafuatayo, kupanua kila binomial, kwa kutumia Theorem ya Binomial.

    1. \((p+q)^{6}\)
    2. \((t-1)^{9}\)
    3. \((2 x+1)^{4}\)
    4. \((4 x+3 y)^{4}\)
    5. \((x-3 y)^{5}\)
    Jibu

    2. \(\begin{array}{l}{t^{9}-9 t^{8}+36 t^{7}-84 t^{6}+126 t^{5}} {-126 t^{4}+84 t^{3}-36 t^{2}+9 t-1}\end{array}\)

    4. \(\begin{array}{l}{256 x^{4}+768 x^{3} y+864 x^{2} y^{2}} {+432 x y^{3}+81 y^{4}}\end{array}\)

    Zoezi\(\PageIndex{26}\) Use the Binomial Theorem to Expand a Binomial

    Katika mazoezi yafuatayo, tafuta neno lililoonyeshwa katika upanuzi wa binomial.

    1. Muda wa saba\((a+b)^{9}\)
    2. Muda wa tatu wa\((x-y)^{7}\)
    Jibu

    1. \(84a^{6} b^{3}\)

    Zoezi\(\PageIndex{27}\) Use the Binomial Theorem to Expand a Binomial

    Katika mazoezi yafuatayo, tafuta mgawo wa muda ulioonyeshwa katika upanuzi wa binomial.

    1. \(y^{4}\)muda wa\((y+3)^{6}\)
    2. \(x^{5}\)muda wa\((x-2)^{8}\)
    3. \(a^{3} b^{4}\)muda wa\((2 a+b)^{7}\)
    Jibu

    1. \(135\)

    3. \(280\)

    Mazoezi mtihani

    Zoezi\(\PageIndex{28}\)

    Katika mazoezi yafuatayo, weka maneno matano ya kwanza ya mlolongo ambao muda wake umetolewa.

    1. \(a_{n}=\frac{5 n-3}{3^{n}}\)
    2. \(a_{n}=\frac{(n+2) !}{(n+3) !}\)
    3. Pata muda wa jumla kwa mlolongo,\(-\frac{2}{3},-\frac{4}{5},-\frac{6}{7},-\frac{8}{9},-\frac{10}{11}, \dots\)
    4. Panua jumla ya sehemu na kupata thamani yake. \(\sum_{i=1}^{4}(-4)^{i}\)
    5. Andika zifuatazo kwa kutumia maelezo ya muhtasari. \(-1+\frac{1}{4}-\frac{1}{9}+\frac{1}{16}-\frac{1}{25}\)
    6. Andika maneno matano ya kwanza ya mlolongo wa hesabu na neno la kwanza na tofauti ya kawaida. \(a_{1}=-13\)na\(d=3\)
    7. Kupata muda wa ishirini ya mlolongo hesabu ambapo muda wa kwanza ni mbili na tofauti ya kawaida ni\(−7\).
    8. Pata muda wa ishirini na tatu wa mlolongo wa hesabu ambao muda wake wa saba ni\(11\) na tofauti ya kawaida ni tatu. Kisha tafuta formula kwa muda mrefu.
    9. Pata neno la kwanza na tofauti ya kawaida ya mlolongo wa hesabu ambao muda wa tisa ni\(−1\) na muda wa kumi na sita ni\(−15\). Kisha tafuta formula kwa muda mrefu.
    10. Pata jumla ya\(25\) masharti ya kwanza ya mlolongo wa hesabu,\(5,9,13,17,21, \dots\)
    11. Pata jumla ya\(50\) maneno ya kwanza ya mlolongo wa hesabu ambao muda wake ni\(a_{n}=-3 n+100\).
    12. Pata jumla. \(\sum_{i=1}^{40}(5 i-21)\)
    Jibu

    2. \(\frac{1}{4}, \frac{1}{5}, \frac{1}{6}, \frac{1}{7}, \frac{1}{8}\)

    4. \(-4+16-64+256=204\)

    6. \(-13,-10,-7,-4,-1\)

    8. \(a_{23}=59 .\)Neno la jumla ni\(a_{n}=3 n-10\).

    10. \(1,325\)

    12. \(3,260\)

    Zoezi\(\PageIndex{29}\)

    Katika mazoezi yafuatayo, onyesha kama mlolongo ni hesabu, jiometri, au wala. Ikiwa hesabu, basi pata tofauti ya kawaida. Ikiwa kijiometri, kisha pata uwiano wa kawaida.

    1. \(14,3,-8,-19,-30,-41, \ldots\)
    2. \(324,108,36,12,4, \frac{4}{3}, \ldots\)
    3. Andika maneno tano ya kwanza ya mlolongo wa kijiometri na uwiano uliopewa kwanza na uwiano wa kawaida. \(a_{1}=6\)na\(r=−2\).
    4. Katika mlolongo wa kijiometri ambao muda wa kwanza na uwiano wa kawaida ni\(a_{1}=5\) na\(r=4\), tafuta\(a_{11}\).
    5. Kupata\(a_{10}\) ya mlolongo wa kijiometri,\(1250,250,50,10,2, \frac{2}{5}, \ldots\) Kisha kupata
      formula kwa muda wa jumla.
    6. Pata jumla ya masharti kumi na tatu ya mlolongo wa kijiometri,\(2,-6,18,-54,162,-486 \ldots\)
    Jibu

    2. Mlolongo ni kijiometri na uwiano wa kawaida\(r=\frac{1}{3}\).

    4. \(5,242,880\)

    6. \(797,162\)

    Zoezi\(\PageIndex{30}\)

    Katika mazoezi yafuatayo, pata jumla.

    1. \(\sum_{i=1}^{9} 5(2)^{i}\)
    2. \(1-\frac{1}{5}+\frac{1}{25}-\frac{1}{125}+\frac{1}{625}-\frac{1}{3125}+\dots\)
    3. Andika decimal kurudia kama sehemu. \(0 . \overline{81}\)
    4. Dave alipata kazi yake ya kwanza ya muda baada ya kuhitimu kutoka shule ya sekondari akiwa na umri wa miaka 18. Aliamua kuwekeza $\(450\) kwa mwezi katika IRA (annuity). Maslahi juu ya annuity\(6\) ni% ambayo imezungukwa kila mwezi. Ni kiasi gani katika akaunti ya Adamu atakapostaafu katika siku yake ya kuzaliwa sitini na tano?
    5. Panua binomial kwa kutumia Triangle ya Pascal. \((m-2 n)^{5}\)
    6. Tathmini kila mgawo wa binomial.
      1. \(\left( \begin{array}{l}{8} \\ {1}\end{array}\right)\)
      2. \(\left( \begin{array}{l}{16} \\ {16}\end{array}\right)\)
      3. \(\left( \begin{array}{l}{12} \\ {0}\end{array}\right)\)
      4. \(\left( \begin{array}{l}{10} \\ {6}\end{array}\right)\)
    7. Panua binomial kwa kutumia Theorem ya Binomial. \((4 x+5 y)^{3}\)
    Jibu

    2. \(\frac{5}{6}\)

    4. \(\$ 1,409,344.19\)

    6.

    1. \(8\)
    2. \(1\)
    3. \(1\)
    4. \(210\)