CONTOH SOAL 2 BARISAN

 $\begin{aligned}3.\quad &(\textbf{LM UGM ke-19 Th.2007 Tk. SMA})\\ &\textrm{Diketahui barisan geometri dengan suku pertama}\quad a\\ &\textrm{dan rasio}\quad r.\:\: \textrm{Untuk sebarang}\quad n\quad \textrm{genap didefinisikan}\\ &\textrm{jumlahan}\quad S_{\displaystyle n}=U_{\displaystyle 1}+U_{\displaystyle 2}+U_{\displaystyle 3}+...+U_{\displaystyle n}\quad \textrm{dan}\\ &\widehat{S}_{\displaystyle n}=U_{\displaystyle 1}-U_{\displaystyle 2}+U_{\displaystyle 3}-U_{\displaystyle 4}+...+U_{\displaystyle n-1}-U_{\displaystyle n}.\quad \textrm{Nilai}\\ &r\quad \textrm{yang mungkin agar}\quad \displaystyle \frac{\widehat{S}_{\displaystyle n}}{S_{\displaystyle n}}>3?\\ &\text{a}.\quad -1< r<0\\ &\text{b}.\quad -1< r<\displaystyle -\frac{1}{2}\\ &\text{c}.\quad -1< r<\displaystyle -\frac{1}{3}\\ &\text{d}.\quad \displaystyle -\frac{1}{3}< r<0\\ &\text{e}.\quad \textrm{tergantung oleh nilai}\quad a\\\\ &\textbf{Jawab:    b}\\ &\begin{aligned}&\textrm{Suku-suku barisan geometri adalah}:\: U_{\displaystyle n}=a.r^{\displaystyle n-1}\\ &\textrm{Untuk}\quad n\quad \textrm{genap, maka}:\\ &\textbf{Menentukan}\quad S_{\displaystyle n}\\ &S_{\displaystyle n}=U_{\displaystyle 1}+U_{\displaystyle 2}+U_{\displaystyle 3}+...+U_{\displaystyle n}=a+ar+ar^{\displaystyle 2}+...+ar^{n-1}\\ &\:\:\:\quad=a\left( \displaystyle \frac{1-r^{\displaystyle n}}{1-r} \right)\\ &\textbf{Menentukan}\quad \widehat{S}_{\displaystyle n}\\ &\widehat{S}_{\displaystyle n}=U_{\displaystyle 1}-U_{\displaystyle 2}+U_{\displaystyle 3}-U_{\displaystyle 4}+...+U_{\displaystyle n-1}-U_{\displaystyle n}\\ &\:\:\quad =a-ar+ar^{\displaystyle 2}+ar^{\displaystyle 4}+...+a.r^{n-1}\\ &\qquad \textrm{Karena}:\: (-r)^{\displaystyle n}=r^{\displaystyle n}\\ &\:\:\:\quad=a\left( \displaystyle \frac{1-r^{\displaystyle n}}{1+r} \right)\\ &\textbf{Menentukan rasio}\\ &\displaystyle \frac{\widehat{S}_{\displaystyle n}}{S_{\displaystyle n}}>3\Leftrightarrow \displaystyle \frac{a\left( \displaystyle \frac{1-r^{\displaystyle n}}{1+r} \right)}{a\displaystyle \left( \frac{1-r^{\displaystyle n}}{1-r} \right)}>3\Leftrightarrow \displaystyle \frac{1-r}{1+r}\gt 3\\ &\Leftrightarrow \displaystyle \frac{1-r}{1+r}-3\gt 0\\ &\Leftrightarrow \displaystyle \frac{1-r-3(r+1)}{1+r}\gt 0\\ &\Leftrightarrow \displaystyle \frac{-4r-2}{1+r}\gt 0\quad (\textrm{masing-masing ruas dikali dengan }-1)\\ &\Leftrightarrow \displaystyle \frac{4r+2}{1+r}\lt  0\\ &\textrm{Secara ketaksamaan wilayah}\quad r\quad \textrm{akan berada di}\\ &-1\lt r\lt \displaystyle -\frac{1}{2}\end{aligned} \end{aligned}$