BARISAN GEOMETRI
Suatu barisan bilangan disebut dengan barisan geometri jika memiliki rasio yang tetap.
Rasio merupakan perbandingan antara dua suku yang berturutan
\(\text{r} = \dfrac{\text{U}_2}{\text{U}_1} = \dfrac{\text{U}_3}{\text{U}_2} = \color{blue}\dfrac{\text{U}_{\text{(n + 1)}}}{\text{U}_{\text{n}}}\)
Contoh:
3, 15, 75, 375, 1875, …
Rasio barisan di atas adalah \(\text{r} = \dfrac{15}{3} = 5\), dan nilai rasio ini selalu konstan (tetap), sehingga barisan di atas merupakan barisan geometri.
Rumus Umum Suku ke-n
\(\text{U}_1, \text{U}_2, \text{U}_3, \text{U}_4, \dotso\)
\(\color{blue}\text{U}_{\text{n}} = \text{a}\cdot \text{r}^{\text{n – 1}}\)
dengan \(\text{a}\) adalah suku pertama dan \(\text{r}\) adalah rasio
Contoh:
Tentukan rumus umum suku ke-n dari barisan geometri berikut:
\(2, -8, 32, -128, \dotso, 8192\)
\(\text{r} = \dfrac{-8}{2} = -4\)
\(\text{U}_{\text{n}} = \text{a}\cdot \text{r}^{\text{n – 1}}\)
\(\text{U}_{\text{n}} = 2\cdot (-4)^{\text{n – 1}}\)
\(\text{U}_{\text{n}} = 2\cdot (-4)^{\text{n}}\cdot (-4)^{-1}\)
\(\text{U}_{\text{n}} = 2\cdot (-4)^{\text{n}}\cdot (-\frac{1}{4})\)
\(\text{U}_{\text{n}} = -\frac{1}{2}\cdot (-4)^{\text{n}}\)
Suku Tengah
\(\color{blue}\text{U}_{\text{t}} = \sqrt{\text{a}\cdot \text{U}_{\text{n}}}\)
DERET GEOMETRI
Rumus Jumlah n suku pertama
\(\text{U}_1 + \text{U}_2 + \text{U}_3 + \text{U}_4 + \dotso\)
\(\color{blue}\text{S}_{\text{n}} = \dfrac{\text{a}(\text{r}^{\text{n}}\:-\:1)}{\text{r}\:-\:1}\)
Contoh:
Tentukan jumlah 10 suku pertama untuk deret geometri berikut:
\(\dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{8} + \dfrac{1}{16} + \dotso\)
\(\text{r} = \dfrac{\frac{1}{4} }{\frac{1}{2}} = \dfrac{2}{4} = \dfrac{1}{2}\)
\(\text{S}_{10} = \dfrac{\frac{1}{2}\left[(\frac{1}{2})^{10}\:-\:1\right])}{\frac{1}{2}\:-\:1}\)
\(\text{S}_{10} = \dfrac{\frac{1}{2}(\frac{1}{1024}\:-\:1)}{-\frac{1}{2}}\)
\(\text{S}_{10} = \dfrac{-\frac{1}{2}\cdot\frac{1023}{1024}}{-\frac{1}{2}}\)
\(\text{S}_{10} = \dfrac{-\frac{1023}{2048}}{-\frac{1}{2}}\)
\(\text{S}_{10} = \dfrac{1023}{2048}\times \dfrac{2}{1}\)
\(\text{S}_{10} = \dfrac{1023}{1024}\)