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12.4E: Mazoezi

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    Mazoezi hufanya kamili

    Zoezi\(\PageIndex{21}\) 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. \(2,10,50,250,1250,6250, \dots\)
    3. \(48,24,12,6,3, \frac{3}{2}, \dots\)
    4. \(54,18,6,2, \frac{2}{3}, \frac{2}{9}, \dots\)
    5. \(-3,6,-12,24,-48,96, \dots\)
    6. \(2,-6,18,-54,162,-486, \dots\)
    Jibu

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

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

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

    Zoezi\(\PageIndex{22}\) Determine if a Sequence is Geometric

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

    1. \(48,24,12,6,3, \frac{3}{2}, \ldots\)
    2. \(12,6,0,-6,-12,-18, \dots\)
    3. \(-7,-2,3,8,13,18, \dots\)
    4. \(5,9,13,17,21,25, \ldots\)
    5. \(\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, \frac{1}{64}, \dots\)
    6. \(4,8,12,24,48,96, \dots\)
    Jibu

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

    3. Mlolongo ni hesabu na tofauti ya kawaida\(d=5\).

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

    Zoezi\(\PageIndex{23}\) 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. \(a_{1}=4\)na\(r=3\)
    2. \(a_{1}=9\)na\(r=2\)
    3. \(a_{1}=-4\)na\(r=-2\)
    4. \(a_{1}=-5\)na\(r=-3\)
    5. \(a_{1}=27\)na\(r=\frac{1}{3}\)
    6. \(a_{1}=64\)na\(r=\frac{1}{4}\)
    Jibu

    1. \(4,12,36,108,324\)

    3. \(-4,8,-16,32,-64\)

    5. \(27,9,3,1, \frac{1}{3}\)

    Zoezi\(\PageIndex{24}\) Find the General Term (\(n\)th Term) of a Geometric Sequence
    1. Kupata\(a_{11}\) aliyopewa\(a_{1}=8\) na\(r=3\).
    2. Kupata\(a_{13}\) aliyopewa\(a_{1}=7\) na\(r=2\).
    3. Kupata\(a_{10}\) aliyopewa\(a_{1}=-6\) na\(r=-2\).
    4. Kupata\(a_{15}\) aliyopewa\(a_{1}=-4\) na\(r=-3\).
    5. Kupata\(a_{10}\) aliyopewa\(a_{1}=100,000\) na\(r=0.1\).
    6. Kupata\(a_{8}\) aliyopewa\(a_{1}=1,000,000\) na\(r=0.01\).
    Jibu

    1. \(472,392\)

    3. \(3,072\)

    5. \(0.0001\)

    Zoezi\(\PageIndex{25}\) Find the General Term (\(n\)th Term) of a Geometric Sequence

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

    1. Kupata\(a_{9}\) ya mlolongo,\(9,18,36,72,144,288, \dots\)
    2. Kupata\(a_{12}\) ya mlolongo,\(5,15,45,135,405,1215, \dots\)
    3. Kupata\(a_{15}\) ya mlolongo,\(-486,162,-54,18,-6,2, \dots\)
    4. Kupata\(a_{16}\) ya mlolongo,\(224,-112,56,-28,14,-7, \ldots\)
    5. Kupata\(a_{10}\) ya mlolongo,\(1,0.1,0.01,0.001,0.0001,0.00001, \ldots\)
    6. Kupata\(a_{9}\) ya mlolongo,\(1000,100,10,1,0.1,0.01, \dots\)
    Jibu

    1. \(a_{9}=2,304 .\)Neno la jumla ni\(a_{n}=9(2)^{n-1}\).

    3. \(a_{15}=-\frac{2}{19,683} .\)Neno la jumla ni\(a_{n}=-486\left(-\frac{1}{3}\right)^{n-1}\).

    5. \(a_{10}=0.000000001 .\)Neno la jumla ni\(a_{n}=(0.1)^{n-1}\).

    Zoezi\(\PageIndex{26}\) 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. \(8,24,72,216,648,1944, \dots\)
    2. \(7,14,28,56,112,224, \dots\)
    3. \(-6,12,-24,48,-96,192, \dots\)
    4. \(-4,12,-36,108,-324,972, \ldots\)
    5. \(81,27,9,3,1, \frac{1}{3}, \ldots\)
    6. \(256,64,16,4,1, \frac{1}{4}, \frac{1}{16}, \dots\)
    Jibu

    1. \(57,395,624\)

    3. \(-65,538\)

    5. \(\frac{7,174,453}{59,049} \approx 121.5\)

    Zoezi\(\PageIndex{27}\) Find the Sum of the First \(n\) terms of a Geometric Sequence

    Katika mazoezi yafuatayo, pata jumla ya mlolongo wa kijiometri.

    1. \(\sum_{i=1}^{15}(2)^{i}\)
    2. \(\sum_{i=1}^{10}(3)^{i}\)
    3. \(\sum_{i=1}^{9} 4(2)^{i}\)
    4. \(\sum_{i=1}^{8} 5(3)^{i}\)
    5. \(\sum_{i=1}^{10} 9\left(\frac{1}{3}\right)^{i}\)
    6. \(\sum_{i=1}^{15} 4\left(\frac{1}{2}\right)^{i}\)
    Jibu

    1. \(65,534\)

    3. \(4088\)

    5. \(\frac{29,524}{6561} \approx 4.5\)

    Zoezi\(\PageIndex{28}\) Find the Sum of an Infinite Geometric Series

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

    1. \(1+\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\frac{1}{81}+\frac{1}{243}+\frac{1}{729}+\ldots\)
    2. \(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\ldots\)
    3. \(6-2+\frac{2}{3}-\frac{2}{9}+\frac{2}{27}-\frac{2}{81}+\ldots\)
    4. \(-4+2-1+\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\dots\)
    5. \(6+12+24+48+96+192+\dots\)
    6. \(5+15+45+135+405+1215+\ldots\)
    7. \(1,024+512+256+128+64+32+\ldots\)
    8. \(6,561+2187+729+243+81+27+\dots\)
    Jibu

    1. \(\frac{3}{2}\)

    3. \(\frac{9}{2}\)

    5. hakuna jumla kama\(r≥1\)

    7.\ (2,048\ (

    Zoezi\(\PageIndex{29}\) Find the Sum of an Infinite Geometric Series

    Katika mazoezi yafuatayo, andika kila decimal kurudia kama sehemu.

    1. \(0 . \overline{3}\)
    2. \(0 . \overline{6}\)
    3. \(0 . \overline{7}\)
    4. \(0 . \overline{2}\)
    5. \(0 . \overline{45}\)
    6. \(0 . \overline{27}\)
    Jibu

    1. \(\frac{1}{3}\)

    3. \(\frac{7}{9}\)

    5. \(\frac{5}{11}\)

    Zoezi\(\PageIndex{30}\) Apply Geometric Sequences and Series in the Real World

    Katika mazoezi yafuatayo, tatua tatizo.

    1. Pata athari ya jumla juu ya uchumi wa kila marupurupu ya kodi ya serikali kwa kila kaya ili kuchochea uchumi ikiwa kila kaya itatumia asilimia iliyoonyeshwa ya marupurupu katika bidhaa na huduma.
    Kodi ya marupurupu kwa kila kaya Asilimia alitumia katika bidhaa na huduma Jumla ya Athari juu ya uchumi
    a. $\(1,000\) \(85\)%  
    b. $\(1,000\) \(75\)%  
    c. $\(1,500\) \(90\)%  
    d. $\(1,500\) \(80\)%  
    Jedwali 12.3.3

    2. New mababu kuamua kuwekeza $\(100\) kwa mwezi katika annuity kwa mjukuu wao. Akaunti italipa riba\(6\)% kwa mwaka ambayo imezungukwa kila mwezi (\(12\)mara kwa mwaka). Kiasi gani kitakuwa katika akaunti ya mtoto wakati wa kuzaliwa kwao ishirini na moja?

    3. Berenice alipata kazi yake ya kwanza ya wakati wote baada ya kuhitimu kutoka chuo kikuu akiwa na umri\(30\). Aliamua kuwekeza $\(500\) kwa robo katika IRA (annuity). Maslahi juu ya annuity\(7\) ni% ambayo imezungukwa robo mwaka (\(4\)mara kwa mwaka). Kiasi gani kitakuwa katika akaunti ya Berenice wakati anastaafu akiwa na umri\(65\)?

    4. Alice anataka kununua nyumba katika miaka mitano. Yeye ni kuweka $\(500\) mwezi katika annuity kwamba chuma\(5\)% kwa mwaka kwamba ni imezungukwa kila mwezi (\(12\)mara kwa mwaka). Kiasi gani Alice na kwa ajili ya malipo yake chini katika miaka mitano?

    5. Myra alipata kazi yake ya kwanza ya wakati wote baada ya kuhitimu kutoka chuo kikuu. Anapanga kupata shahada ya bwana, na hivyo ni kuweka $ mwaka kutoka\(2,500\) kwa bonus yake ya mwisho wa mwaka katika annuity. Annuity inalipa\(6.5\)% kwa mwaka na imezungukwa kila mwaka. Je! Ataokoa kiasi gani katika miaka mitano ili afuate shahada yake ya bwana?

    Jibu

    1. a. $\(6666.67\) b. $\(4000\) c. $\(15,000\) d. $\(7500\)

    3. $\(295,581.88\)

    5. $\(14,234.10\)

    Zoezi\(\PageIndex{31}\) Writing Exercises
    1. Kwa maneno yako mwenyewe, kuelezea jinsi ya kuamua kama mlolongo ni kijiometri.
    2. Kwa maneno yako mwenyewe, kuelezea jinsi ya kupata muda wa jumla wa mlolongo wa kijiometri.
    3. Kwa maneno yako mwenyewe, kuelezea tofauti kati ya mlolongo wa kijiometri na mfululizo wa kijiometri.
    4. Kwa maneno yako mwenyewe, kuelezea jinsi ya kuamua kama mfululizo usio na kipimo wa kijiometri una jumla na jinsi ya kuipata.
    Jibu

    2. Majibu yatatofautiana.

    4. Majibu yatatofautiana

    Self Check

    Takwimu hii inaonyesha safu saba na nguzo nne. Mstari wa kwanza ni mstari wa kichwa na unasoma, “Naweza”, “Kwa ujasiri”, “Kwa msaada fulani”, na “Hapana, siipati. Safu ya kwanza inasoma, “Tambua kama mlolongo ni kijiometri”, “Pata neno la jumla (neno la nth) la”, “Mlolongo wa kijiometri”, “Tafuta jumla ya mfululizo wa kijiometri usio”, Tumia utaratibu wa kijiometri kutatua programu”. Nguzo zilizobaki ni tupu.
    Kielelezo 12.3.11
    1. Baada ya kukamilisha mazoezi, tumia orodha hii ili kutathmini ujuzi wako wa malengo ya sehemu hii.
    2. Orodha hii inakuambia nini kuhusu ujuzi wako wa sehemu hii? Ni hatua gani utachukua ili kuboresha?