Lanjutan 2 Contoh Soal Barisan dan Deret

$\begin{array}{l}\\ 6.& (\mathbf{\text{EBTANAS 2000}})\\ & \text{Diketahui}{\displaystyle \sum _{k=5}^{25}(2-pk)=0,\text{maka nilai}}\\ & \sum _{k=5}^{25}pk=....\\ & \text{A}.20\text{D}.42\\ & \text{B}.28\text{C}.30\text{E}.112\\ \\ & \mathbf{\text{Jawab}}:\\ & \begin{array}{rl}{\displaystyle \sum _{k=5}^{25}(2-pk)={\displaystyle \sum _{k=5}^{25}2-\sum _{k=5}^{25}pk}}& =0\\ {\displaystyle \sum _{k=5-4}^{25-4}2-\sum _{k=5}^{25}pk}& =0\\ {\displaystyle \sum _{k=5}^{25}pk}& ={\displaystyle \sum _{k=1}^{21}2}\\ & =21.2\\ & =42\end{array}\end{array}$.

$\begin{array}{l}\\ 7.& (\mathbf{\text{EBTANAS 2000}})\\ & \text{Nilai dari}{\displaystyle \sum _{k=1}^7{\left({\displaystyle \frac{1}{2}}\right)}^{k+1}=....}\\ & \text{A}.{\displaystyle \frac{127}{1024}\text{D}.{\displaystyle \frac{127}{128}}}\\ \\ & \text{B}.{\displaystyle \frac{127}{256}\text{C}.{\displaystyle \frac{255}{512}\text{E}.{\displaystyle \frac{255}{256}}}}\\ \\ & \mathbf{\text{Jawab}}:\\ & \begin{array}{rl}& {\displaystyle \sum _{k=1}^7{\left({\displaystyle \frac{1}{2}}\right)}^{k+1}}\\ & ={\left({\displaystyle \frac{1}{2}}\right)}^2+{\left(\frac{1}{2}\right)}^3+{\left(\frac{1}{2}\right)}^4+{\left(\frac{1}{2}\right)}^5+{\left(\frac{1}{2}\right)}^6+{\left(\frac{1}{2}\right)}^7+{\left({\displaystyle \frac{1}{2}}\right)}^8\\ & ={\displaystyle \frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}+\frac{1}{256}}\\ & ={\displaystyle \frac{64+32+16+8+4+2+1}{256}}\\ & ={\displaystyle \frac{127}{256}}\end{array}\end{array}$.

$\begin{array}{l}\\ 8.& (\mathbf{\text{UN 2013}})\\ & \text{Diketahui suatu barisan aritmetika dengan suku }\\ & \text{ketiga adalah 4 dan suku ketujuhnya adalah 16}.\\ & \text{Jumlah 10 suku pertama dari deret tersebut }\\ & \text{adalah}....\\ & \text{A}.115\text{D}.135\\ & \text{B}.125\text{C}.130\text{E}.{\displaystyle 140}\\ \\ & \mathbf{\text{Jawab}}:\\ & \text{Dari soal diketahui bahwa}\\ & \begin{array}{rl}{U}_3=a+2b& =4\\ U_7=a+6b& =16{}_{-}\\ ------& --\\ -4b& =-12\\ b& =3\\ \text{Sehingga}& \text{didapatkan}\\ \text{nilai}a=& 4-2b\\ =& 4-2.3\\ =& -2\end{array}\\ & \begin{array}{rl}\text{Maka}& \text{jumlah 10 suku pertama deret tersebut adalah}\\ S_n& ={\displaystyle \frac{n}{2}(2a+(n-1)b)}\\ S_{10}& ={\displaystyle \frac{10}{2}(2.(-2)+(10-1).3)}\\ & =5(-4+27)\\ & =5(23)\\ & =115\end{array}\end{array}$.

$\begin{array}{l}\\ 9.& (\mathbf{\text{UN 2014}})\\ & \text{Diketahui tempat duduk gedung pertunjukan}\\ & \text{film diatur mulai dari baris depan ke belakang }\\ & \text{dengan banyak banyak baris dibelakang lebih }\\ & \text{4 kursi dari baris di depannya. Jika dalam }\\ & \text{gedung pertunjukan terdapat 15 baris kursi dan}\\ & \text{baris terdepan ada 20 kursi, maka kapasitas }\\ & \text{gedung pertunjukan tersebut adalah}....\text{kursi}\\ & \text{A}.1200\text{D}.600\\ & \text{B}.800\text{C}.720\text{E}.300\\ \\ & \mathbf{\text{Jawab}}:\\ & \begin{array}{rl}\text{Diketahui}& \{\begin{array}{ll}a& =U_1=20\\ b& =4\\ n& =15\\ S_n& ={\displaystyle \frac{n}{2}(2a+(n-1)b)}\end{array}\end{array}\\ & \begin{array}{rl}{S}_n& ={\displaystyle \frac{n}{2}(2a+(n-1)b)}\\ & ={\displaystyle \frac{15}{2}(2(20)+(15-1).4)}\\ & =15(20+28)\\ & =15(48)\\ & =750\end{array}\end{array}$.

$\begin{array}{l}\\ 10.& (\mathbf{\text{UN 2015}})\\ & \text{Diketahui suatu barisan aritmetika dengan suku}\\ & \text{ke-3 adalah 2 dan suku ke-8 adalah -13. Jumlah}\\ & \text{20 suku pertama dari deret tersebut adalah}....\\ & \text{A}.-580\text{D}.-410\\ & \text{B}.-490\text{C}.-440\text{E}.-380\\ \\ & \mathbf{\text{Jawab}}:\\ & \text{Dari soal di atas diketahui bahwa}\\ & \begin{array}{rl}{U}_3=a+2b& =2\\ U_8=a+7b& =-13{}_{-}\\ ------& ---\\ -5b& =15\\ b& =-3\\ \text{Sehingga}& \text{didapatkan}\\ \text{nilai}a=& 2-2b\\ =& 2-2.(-3)\\ =& 8\end{array}\\ & \begin{array}{rl}\text{Maka}& \text{jumlah 20 suku pertama deret tersebut adalah}\\ S_n& ={\displaystyle \frac{n}{2}(2a+(n-1)b)}\\ S_{20}& ={\displaystyle \frac{20}{2}(2.(8)+(20-1).(-3))}\\ & =10(16-57)\\ & =10(-41)\\ & =-410\end{array}\end{array}$.