12.3E: Mazoezi
- Page ID
- 175711
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\dsum}{\displaystyle\sum\limits} \)
\( \newcommand{\dint}{\displaystyle\int\limits} \)
\( \newcommand{\dlim}{\displaystyle\lim\limits} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\(\newcommand{\longvect}{\overrightarrow}\)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Mazoezi hufanya kamili
Katika mazoezi yafuatayo, onyesha kama kila mlolongo ni hesabu, na ikiwa ni hivyo, onyesha tofauti ya kawaida.
- \(4,12,20,28,36,44, \dots\)
- \(-7,-2,3,8,13,18, \dots\)
- \(-15,-16,3,12,21,30, \dots\)
- \(11,5,-1,-7-13,-19, \dots\)
- \(8,5,2,-1,-4,-7, \dots\)
- \(15,5,-5,-15,-25,-35, \dots\)
- Jibu
-
1. Mlolongo ni hesabu na tofauti ya kawaida\(d=8\).
3. Mlolongo sio hesabu.
5. Mlolongo ni hesabu na tofauti ya kawaida\(d=−3\).
Katika mazoezi yafuatayo, weka maneno matano ya kwanza ya kila mlolongo na neno la kwanza na tofauti ya kawaida.
- \(a_{1}=11\)na\(d=7\)
- \(a_{1}=18\)na\(d=9\)
- \(a_{1}=-7\)na\(d=4\)
- \(a_{1}=-8\)na\(d=5\)
- \(a_{1}=14\)na\(d=-9\)
- \(a_{1}=-3\)na\(d=-3\)
- Jibu
-
1. \(11,18,25,32,39\)
3. \(-7,-3,1,5,9\)
5. \(14,5,-4,-13,-22\)
Katika mazoezi yafuatayo, tafuta neno lililoelezwa kwa kutumia habari iliyotolewa.
- Pata muda wa ishirini na moja wa mlolongo ambapo muda wa kwanza ni wa tatu na tofauti ya kawaida ni nane.
- Pata muda wa ishirini na tatu wa mlolongo ambapo muhula wa kwanza ni sita na tofauti ya kawaida ni nne.
- Pata muda wa thelathini ya mlolongo ambapo muda wa kwanza ni\(−14\) na tofauti ya kawaida ni tano.
- Pata muda wa arobaini wa mlolongo ambapo muda wa kwanza ni\(−19\) na tofauti ya kawaida ni saba.
- Kupata kumi na sita mrefu ya mlolongo ambapo muda wa kwanza ni\(11\) na tofauti ya kawaida ni\(−6\).
- Pata muda wa kumi na nne wa mlolongo ambapo muda wa kwanza ni nane na tofauti ya kawaida ni\(−3\).
- Kupata muda wa ishirini ya mlolongo ambapo muda wa tano ni\(−4\) na tofauti ya kawaida ni\(−2\). Kutoa formula kwa muda wa jumla.
- Pata muda wa kumi na tatu wa mlolongo ambapo muda wa sita ni\(−1\) na tofauti ya kawaida ni\(−4\). Kutoa formula kwa muda wa jumla.
- Kupata kumi na moja mrefu ya mlolongo ambapo muda wa tatu ni\(19\) na tofauti ya kawaida ni tano. Kutoa formula kwa muda wa jumla.
- Pata muda wa kumi na tano wa mlolongo ambapo muda wa kumi ni\(17\) na tofauti ya kawaida ni saba. Kutoa formula kwa muda wa jumla.
- Pata muda wa nane wa mlolongo ambapo muda wa saba ni\(−8\) na tofauti ya kawaida ni\(−5\). Kutoa formula kwa muda wa jumla.
- Kupata kumi na tano mrefu ya mlolongo ambapo mrefu kumi ni\(−11\) na tofauti ya kawaida ni\(−3\). Kutoa formula kwa muda wa jumla.
- Jibu
-
1. \(163\)
3. \(131\)
5. \(-79\)
7. \(a_{20}=-34 .\)Neno la jumla ni\(a_{n}=-2 n+6\).
9. \(a_{11}=59 .\)Neno la jumla ni\(a_{n}=5 n+4\).
11. \(a_{8}=-13 .\)Neno la jumla ni\(a_{n}=-5 n+27\).
Katika mazoezi yafuatayo, tafuta neno la kwanza na tofauti ya kawaida ya mlolongo na masharti yaliyotolewa. Kutoa formula kwa muda wa jumla.
- Muda wa pili ni\(14\) na muda wa kumi na tatu ni\(47\).
- Muda wa tatu ni\(18\) na muda wa kumi na nne ni\(73\).
- Muda wa pili ni\(13\) na muda wa kumi ni\(−51\).
- Muda wa tatu ni nne na mrefu wa kumi ni\(−38\).
- Muda wa nne ni\(−6\) na muda wa kumi na tano ni\(27\).
- Muda wa tatu ni\(−13\) na muda wa kumi na saba ni\(15\).
- Jibu
-
1. \(a_{1}=11, d=3 .\)Neno la jumla ni\(a_{n}=3 n+8\).
3. \(a_{1}=21, d=-8 .\)Neno la jumla ni\(a_{n}=-8 n+29\)
5. \(a_{1}=-15, d=3 .\)Neno la jumla ni\(a_{n}=3 n-18\).
Katika mazoezi yafuatayo, pata jumla ya\(30\) maneno ya kwanza ya kila mlolongo wa hesabu.
- \(11,14,17,20,23, \dots\)
- \(12,18,24,30,36, \dots\)
- \(8,5,2,-1,-4, \dots\)
- \(16,10,4,-2,-8, \dots\)
- \(-17,-15,-13,-11,-9, \dots\)
- \(-15,-12,-9,-6,-3, \dots\)
- Jibu
-
1. \(1,635\)
3. \(-1,065\)
5. \(360\)
Katika mazoezi yafuatayo, tafuta jumla ya\(50\) maneno ya kwanza ya mlolongo wa hesabu ambao muda wake umetolewa.
- \(a_{n}=5 n-1\)
- \(a_{n}=2 n+7\)
- \(a_{n}=-3 n+5\)
- \(a_{n}=-4 n+3\)
- Jibu
-
1. \(6,325\)
3. \(-3,575\)
Katika mazoezi yafuatayo, tafuta kila jumla.
- \(\sum_{i=1}^{40}(8 i-7)\)
- \(\sum_{i=1}^{45}(7 i-5)\)
- \(\sum_{i=1}^{50}(3 i+6)\)
- \(\sum_{i=1}^{25}(4 i+3)\)
- \(\sum_{i=1}^{35}(-6 i-2)\)
- \(\sum_{i=1}^{30}(-5 i+1)\)
- Jibu
-
1. \(6,280\)
3. \(4,125\)
5. \(-3,580\)
- Kwa maneno yako mwenyewe, kuelezea jinsi ya kuamua kama mlolongo ni hesabu.
- Kwa maneno yako mwenyewe, eleza jinsi maneno mawili ya kwanza yanatumiwa kupata muda wa kumi. Onyesha mfano wa kuonyesha maelezo yako.
- Kwa maneno yako mwenyewe, kuelezea jinsi ya kupata muda wa jumla wa mlolongo wa hesabu.
- Kwa maneno yako mwenyewe, kuelezea jinsi ya kupata jumla ya\(n\) maneno ya kwanza ya mlolongo wa hesabu bila kuongeza maneno yote.
- Jibu
-
1. Jibu inaweza kutofautiana
3. Jibu inaweza kutofautiana
Self Check
Baada ya kukamilisha mazoezi, tumia orodha hii ili kutathmini ujuzi wako wa malengo ya sehemu hii.
b Baada ya kuchunguza orodha hii, utafanya nini ili uwe na ujasiri kwa malengo yote?