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12.2E: Mazoezi

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    Mazoezi hufanya kamili

    Zoezi\(\PageIndex{25}\) Write the First Few Terms of a Sequence

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

    1. \(a_{n}=2 n-7\)
    2. \(a_{n}=5 n-1\)
    3. \(a_{n}=3 n+1\)
    4. \(a_{n}=4 n+2\)
    5. \(a_{n}=2^{n}+3\)
    6. \(a_{n}=3^{n}-1\)
    7. \(a_{n}=3^{n}-2 n\)
    8. \(a_{n}=2^{n}-3 n\)
    9. \(a_{n}=\frac{2^{n}}{n^{2}}\)
    10. \(a_{n}=\frac{3^{n}}{n^{3}}\)
    11. \(a_{n}=\frac{4 n-2}{2^{n}}\)
    12. \(a_{n}=\frac{3 n+3}{3^{n}}\)
    13. \(a_{n}=(-1)^{n} \cdot 2 n\)
    14. \(a_{n}=(-1)^{n} \cdot 3 n\)
    15. \(a_{n}=(-1)^{n+1} n^{2}\)
    16. \(a_{n}=(-1)^{n+1} n^{4}\)
    17. \(a_{n}=\frac{(-1)^{n+1}}{n^{2}}\)
    18. \(a_{n}=\frac{(-1)^{n+1}}{2 n}\)
    Jibu

    1. \(-5,-3,-1,1,3\)

    3. \(4,7,10,13,16\)

    5. \(5,7,11,19,35\)

    7. \(1,5,21,73,233\)

    9. \(2,1, \frac{8}{9}, 1, \frac{32}{25}\)

    11. \(1, \frac{3}{2}, \frac{5}{4}, \frac{7}{8}, \frac{9}{16}\)

    13. \(-2,4,-6,8,-10\)

    15. \(1,-4,9,-16,25\)

    17. \(1,-\frac{1}{4}, \frac{1}{9},-\frac{1}{16}, \frac{1}{25}\)

    Zoezi\(\PageIndex{26}\) 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. \(8,16,24,32,40, \dots\)
    2. \(7,14,21,28,35, \ldots\)
    3. \(6,7,8,9,10, \dots\)
    4. \(-3,-2,-1,0,1, \dots\)
    5. \(e^{3}, e^{4}, e^{5}, e^{6}, e^{7}, \ldots\)
    6. \(\frac{1}{e^{2}}, \frac{1}{e}, 1, e, e^{2}, \ldots\)
    7. \(-5,10,-15,20,-25, \dots\)
    8. \(-6,11,-16,21,-26, \dots\)
    9. \(-1,8,-27,64,-125, \dots\)
    10. \(2,-5,10,-17,26, \dots\)
    11. \(-2,4,-6,8,-10, \dots\)
    12. \(1,-3,5,-7,9, \dots\)
    13. \(\frac{1}{4}, \frac{1}{16}, \frac{1}{64}, \frac{1}{256}, \frac{1}{1,024}, \dots\)
    14. \(\frac{1}{1}, \frac{1}{8}, \frac{1}{27}, \frac{1}{64}, \frac{1}{125}, \dots\)
    15. \(-\frac{1}{2},-\frac{2}{3},-\frac{3}{4},-\frac{4}{5},-\frac{5}{6}, \dots\)
    16. \(-2,-\frac{3}{2},-\frac{4}{3},-\frac{5}{4},-\frac{6}{5}, \dots\)
    17. \(-\frac{5}{2},-\frac{5}{4},-\frac{5}{8},-\frac{5}{16},-\frac{5}{32}, \dots\)
    18. \(4, \frac{1}{2}, \frac{4}{27}, \frac{4}{64}, \frac{4}{125}, \dots\)
    Jibu

    1. \(a_{n}=8 n\)

    3. \(a_{n}=n+5\)

    5. \(a_{n}=e^{n+2}\)

    7. \(a_{n}=(-1)^{n} 5 n\)

    9. \(a_{n}=(-1)^{n} n^{3}\)

    11. \(a_{n}=(-1)^{n} 2 n\)

    13. \(a_{n}=\frac{1}{4^{n}}\)

    15. \(a_{n}=-\frac{n}{n+1}\)

    17. \(-\frac{5}{2^{n}}\)

    Zoezi\(\PageIndex{27}\) Use Factorial Notation

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

    1. \(a_{n}=\frac{4}{n !}\)
    2. \(a_{n}=\frac{5}{n !}\)
    3. \(a_{n}=3 n !\)
    4. \(a_{n}=2 n !\)
    5. \(a_{n}=(2 n) !\)
    6. \(a_{n}=(3 n) !\)
    7. \(a_{n}=\frac{(n-1) !}{(n) !}\)
    8. \(a_{n}=\frac{n !}{(n+1) !}\)
    9. \(a_{n}=\frac{n !}{n^{2}}\)
    10. \(a_{n}=\frac{n^{2}}{n !}\)
    11. \(a_{n}=\frac{(n+1) !}{n^{2}}\)
    12. \(a_{n}=\frac{(n+1) !}{2 n}\)
    Jibu

    1. \(4,2, \frac{2}{3}, \frac{1}{6}, \frac{1}{30}\)

    3. \(3,6,18,72,360\)

    5. \(2,24,720,40320,3628800\)

    7. \(1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}\)

    9. \(1, \frac{1}{2}, \frac{2}{3}, \frac{3}{2}, \frac{24}{5}\)

    11. \(2, \frac{3}{2}, \frac{8}{3}, \frac{15}{2}, \frac{144}{5}\)

    Zoezi\(\PageIndex{28}\) Find the Partial Sum

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

    1. \(\sum_{i=1}^{5} i^{2}\)
    2. \(\sum_{i=1}^{5} i^{3}\)
    3. \(\sum_{i=1}^{6}(2 i+3)\)
    4. \(\sum_{i=1}^{6}(3 i-2)\)
    5. \(\sum_{i=1}^{4} 2^{i}\)
    6. \(\sum_{i=1}^{4} 3^{i}\)
    7. \(\sum_{k=0}^{3} \frac{4}{k !}\)
    8. \(\sum_{k=0}^{4}-\frac{1}{k !}\)
    9. \(\sum_{k=1}^{5} k(k+1)\)
    10. \(\sum_{k=1}^{5} k(2 k-3)\)
    11. \(\sum_{n=1}^{5} \frac{n}{n+1}\)
    12. \(\sum_{n=1}^{4} \frac{n}{n+2}\)
    Jibu

    1. \(1+4+9+16+25=55\)

    3. \(5+7+9+11+13+15=60\)

    5. \(2+4+8+16=30\)

    7. \(\frac{4}{1}+\frac{4}{1}+\frac{4}{2}+\frac{4}{6}+\frac{32}{3}=10 \frac{2}{3}\)

    9. \(2+6+12+20+30=70\)

    11. \(\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\frac{4}{5}+\frac{5}{6}=\frac{71}{20}\)

    Zoezi\(\PageIndex{29}\) 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. \(\frac{1}{4}+\frac{1}{16}+\frac{1}{64}+\frac{1}{256}\)
    3. \(1+\frac{1}{8}+\frac{1}{27}+\frac{1}{64}+\frac{1}{125}\)
    4. \(\frac{1}{5}+\frac{1}{25}+\frac{1}{125}+\frac{1}{625}\)
    5. \(2+1+\frac{2}{3}+\frac{1}{2}+\frac{2}{5}\)
    6. \(3+\frac{3}{2}+1+\frac{3}{4}+\frac{3}{5}+\frac{1}{2}\)
    7. \(3-6+9-12+15\)
    8. \(-5+10-15+20-25\)
    9. \(-2+4-6+8-10+\ldots+20\)
    10. \(1-3+5-7+9+\ldots+21\)
    11. \(14+16+18+20+22+24+26\)
    12. \(9+11+13+15+17+19+21\)
    Jibu

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

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

    5. \(\sum_{n=1}^{5} \frac{2}{n}\)

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

    9. \(\sum_{n=1}^{10}(-1)^{n} 2 n\)

    11. \(\sum_{n=1}^{7}(2 n+12)\)

    Zoezi\(\PageIndex{30}\) Writing Exercises
    1. Kwa maneno yako mwenyewe, kuelezea jinsi ya kuandika maneno ya mlolongo unapojua formula. Onyesha mfano wa kuonyesha maelezo yako.
    2. Ni maneno gani ya mlolongo ni hasi wakati\(n^{th}\) muda wa mlolongo ni\(a_{n}=(-1)^{n}(n+2)\)?
    3. Kwa maneno yako mwenyewe, kueleza nini maana ya\(n!\) Onyesha baadhi ya mifano kuonyesha maelezo yako.
    4. Eleza nini kila sehemu ya notation\(\sum_{k=1}^{12} 2 k\) ina maana.
    Jibu

    1. Majibu yatatofautiana.

    3. Majibu yatatofautiana.

    Self Check

    Baada ya kukamilisha mazoezi, tumia orodha hii ili kutathmini ujuzi wako wa malengo ya sehemu hii.

    Takwimu hii inaonyesha meza yenye nguzo nne na safu sita. Mstari wa kwanza ni mstari wa kichwa na huandika kila safu, “Siwezi”, “Kwa ujasiri”, “Kwa msaada fulani”, na “Hapana siipati!”. Mstari wa kwanza katika safu ya pili inasoma, “Andika maneno machache ya mlolongo”, mstari wa tatu, safu ya kwanza inasoma, “Tafuta Mfumo wa Muda wa Nth wa Mlolongo”, safu ya nne safu ya kwanza inasoma “Tumia Notation ya Kielelezo, mstari wa tano, safu ya kwanza inasoma, Pata jumla ya sehemu”, na mstari wa mwisho, kwanza safu anayesoma, “Matumizi Summation Notation kuandika Sum”. Nguzo tatu zilizobaki na safu ni tupu.
    Kielelezo 12.1.24

    b Kama wengi wa hundi yako walikuwa:

    ... kwa ujasiri. Hongera! Umefanikiwa malengo katika sehemu hii. Fikiria ujuzi wa kujifunza uliyotumia ili uweze kuendelea kuitumia. Ulifanya nini ili uwe na ujasiri wa uwezo wako wa kufanya mambo haya? Kuwa maalum.

    ... kwa msaada fulani. Hii lazima kushughulikiwa haraka kwa sababu mada huna bwana kuwa mashimo katika barabara yako ya mafanikio. Katika hesabu, kila mada hujenga juu ya kazi ya awali. Ni muhimu kuhakikisha kuwa na msingi imara kabla ya kuendelea. Nani unaweza kuomba msaada? Washiriki wenzako na mwalimu ni rasilimali nzuri. Je, kuna mahali kwenye chuo ambapo waalimu hisabati zinapatikana? Je, ujuzi wako wa kujifunza unaweza kuboreshwa?

    ... hapana - Siipati! Hii ni ishara ya onyo na haipaswi kupuuza. Unapaswa kupata msaada mara moja au utazidiwa haraka. Angalia mwalimu wako haraka iwezekanavyo kujadili hali yako. Pamoja unaweza kuja na mpango wa kupata msaada unayohitaji.