BARISAN ARITMETIKA
Barisan aritmetika merupakan barisan bilangan yang memiliki beda (selisih antara dua suku yang berturutan) yang tetap.
\(\color{blue} \text{b} = \text{U}_{\text{n}}\:-\:\text{U}_{\text{n – 1}}\)
\(\text{b} = \text{U}_{2}\:-\:\text{U}_{1}\)
\(\text{b} = \text{U}_{3}\:-\:\text{U}_{2}\)
Contoh:
3, 9, 15, 21, 27, …
\(\text{b} = 9\:-\:3 = 6\)
Barisan bilangan di atas merupakan barisan aritmetika karena memiliki beda yang tetap yaitu 6.
Rumus Umum Suku ke-n
\(\text{U}_1, \text{U}_2, \text{U}_3, \text{U}_4, \dotso, \text{U}_{\text{n}}\)
\(\text{U}_{\text{n}} = \text{a} + (\text{n}\:-\:1)\text{b}\)
dengan \(\text{a}\) adalah suku pertama dan \(\text{b}\) adalah beda
Contoh:
Tentukan rumus umum suku ke-n dan suku ke-100 dari barisan aritmetika berikut:
\(\frac{2}{5}, \:\frac{3}{5}, \:\frac{4}{5}, \:\frac{5}{5}, \dotso\)
\(\text{b} = \frac{3}{5}\:-\:\frac{2}{5} = \frac{1}{5}\)
\(\text{U}_{\text{n}} = \text{a} + (\text{n}\:-\:1)\text{b}\)
\(\text{U}_{\text{n}} = \frac{2}{5} + (\text{n}\:-\:1)\frac{1}{5}\)
\(\text{U}_{\text{n}} = \frac{2}{5} + \frac{1}{5}\text{n}\:-\:\frac{1}{5}\)
\(\text{U}_{\text{n}} = \frac{1}{5} + \frac{1}{5}\text{n}\)
\(\text{U}_{100} = \frac{1}{5} + \frac{1}{5}\cdot 100\)
\(\text{U}_{100} = \frac{1}{5} + 20\)
\(\text{U}_{100} = 20\frac{1}{5}\)
Suku Tengah
\(\text{U}_t = \dfrac{\text{a} + \text{U}_{\text{n}}}{2}\)
DERET ARITMETIKA
\(\text{U}_1 + \text{U}_2 + \text{U}_3 + \text{U}_4 + \dotso\)
\(\text{S}_{\text{n}} = \dfrac{\text{n}}{2}(\text{a} + \text{U}_{\text{n}})\)
\(\text{S}_{\text{n}} = \dfrac{\text{n}}{2}(\text{a} + \text{a} + (\text{n}\:-\:1)\text{b})\)
\(\color{blue}\text{S}_{\text{n}} = \dfrac{\text{n}}{2}(2\text{a} + (\text{n}\:-\:1)\text{b})\)
Contoh:
Tentukan jumlah 10 suku pertama deret aritmetika berikut:
\(80, 74, 68, 62, \dotso\)
\(\text{b} =74\:-\:80= -6\)
\(\color{blue}\text{S}_{\text{n}} = \dfrac{\text{n}}{2}(2\text{a} + (\text{n}\:-\:1)\text{b})\)
\(\text{S}_{10} = \dfrac{10}{2}[2(80) + (10\:-\:1)(-6)]\)
\(\text{S}_{10} = 5[160 + (9)(-6)]\)
\(\text{S}_{10} = 5(160\:-\:54)\)
\(\text{S}_{10} = 5(106)\)
\(\text{S}_{10} = 530\)