Berapa nilai dari penjumlahan deret di bawah ini?
[tex]\displaystyle \frac{1}{2+\sqrt{2}}+\frac{1}{2\sqrt{3}+3\sqrt{2}}+\frac{1}{6+4\sqrt{3}}+\cdots+\frac{1}{990+100\sqrt{99}}[/tex]
[tex]\displaystyle \frac{1}{2+\sqrt{2}}+\frac{1}{2\sqrt{3}+3\sqrt{2}}+\frac{1}{6+4\sqrt{3}}+\cdots+\frac{1}{990+100\sqrt{99}}[/tex]
Verified answer
Jawab:
[tex]\displaystyle=\bf\frac{9}{10}[/tex]
Penjelasan:
Agak susah nentuin polanya
[tex]\displaystyle\rightarrow\frac{1}{2+\sqrt{2}}+\frac{1}{2\sqrt{3}+3\sqrt{2}}+\frac{1}{6+4\sqrt{3}}+....+\frac{1}{990+100\sqrt{99}}\\\\=\frac{1}{2+\sqrt{2}}+\frac{1}{2\sqrt{3}+3\sqrt{2}}+\frac{1}{6+4\sqrt{3}}+....+\frac{1}{990+100\sqrt{9}\sqrt{11}}\\\\=\frac{1}{2+\sqrt{2}}+\frac{1}{2\sqrt{3}+3\sqrt{2}}+\frac{1}{6+4\sqrt{3}}+....+\frac{1}{990+100(3)\sqrt{11}}\\\\=\frac{1}{2+\sqrt{2}}+\frac{1}{2\sqrt{3}+3\sqrt{2}}+\frac{1}{6+4\sqrt{3}}+....+\frac{1}{990+300\sqrt{11}}[/tex]
[tex]\displaystyle=\frac{1}{2+\sqrt{2}}+\frac{1}{2\sqrt{3}+3\sqrt{2}}+\frac{1}{2(3+2\sqrt{3})}+....+\frac{1}{30(33+10\sqrt{11})}\\\\=\frac{2-\sqrt{2}}{2^2-(\sqrt{2})^2}+\frac{2\sqrt{3}-3\sqrt{2}}{(2\sqrt{3})^2-(3\sqrt{2})^2}+\frac{3-2\sqrt{3}}{2(3^2-(2\sqrt{3})^2)}+....+\frac{33-10\sqrt{11}}{30(33^2-(10\sqrt{11})^2)}\\\\=\frac{2-\sqrt{2}}{4-2}+\frac{2\sqrt{3}-3\sqrt{2}}{12-18}+\frac{3-2\sqrt{3}}{2(9-12)}+....+\frac{33-10\sqrt{11}}{30(1089-1100)}[/tex]
[tex]\displaystyle=\frac{2-\sqrt{2}}{2}-\frac{2\sqrt{3}-3\sqrt{2}}{6}-\frac{3-2\sqrt{3}}{2(3)}+....-\frac{33-10\sqrt{11}}{30(11)}\\\\=\frac{2-\sqrt{2}}{2}-\frac{2\sqrt{3}-3\sqrt{2}}{6}-\frac{3-2\sqrt{3}}{6}+....-\frac{33-10\sqrt{11}}{330}[/tex]
KPK penyebut KPK(2, 6, 330) = 330, oleh karena itu
[tex]\displaystyle=\frac{165(2-\sqrt{2})}{165(2)}-\frac{55(2\sqrt{3}-3\sqrt{2})}{55(6)}-\frac{55(3-2\sqrt{3})}{55(6)}+....-\frac{33-10\sqrt{11}}{330}\\\\=\frac{330-165\sqrt{2}}{165(2)}-\frac{110\sqrt{3}-165\sqrt{2}}{55(6)}-\frac{165-110\sqrt{3}}{55(6)}+....+\frac{10\sqrt{11}-33}{330}\\\\=\frac{330-165\sqrt{2}-110\sqrt{3}+165\sqrt{2}-165+110\sqrt{3}+....+10\sqrt{11}-33}{330}[/tex]
Polanya mulai kelihatan, pindahkan suku-suku pembilang
zig zag zig zag
[tex]\displaystyle=\frac{330-165\sqrt{2}+165\sqrt{2}-110\sqrt{3}+110\sqrt{3}-165+....+10\sqrt{11}-33}{330}\\\\=\frac{330+(-165+....+10\sqrt{11})-33}{330}[/tex]
bagian didalam kurung akan habis ditambah lawannya, maka
[tex]\displaystyle=\frac{330+(-33)}{330}\\\\=\frac{330-33}{330}\\\\=\frac{33(10-1)}{33(10)}\\\\=\frac{\not33(9)}{\not33(10)}[/tex]
[tex]\displaystyle=\bf\frac{9}{10}[/tex]
(xcvi)