Soal yang Akan Dibahas
Nilai dari
$ \frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}} + ... + \frac{1}{\sqrt{63}+\sqrt{64}} = ..... $
A). $ 10 \, $ B). $ 9 \, $ C). $ 8 \, $ D). $ 7 \, $ E). $ 6 $
$ \frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}} + ... + \frac{1}{\sqrt{63}+\sqrt{64}} = ..... $
A). $ 10 \, $ B). $ 9 \, $ C). $ 8 \, $ D). $ 7 \, $ E). $ 6 $
$\spadesuit $ Konsep Dasar
*). Untuk merasionalkan bentuk akar, caranya cukup kali bentuk sekawannya.
bentuk $ \sqrt{a} + \sqrt{b} $ sekawannya adalah $ \sqrt{a} - \sqrt{b} $
Perkaliannya : $ (\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) = a - b $
*). Untuk merasionalkan bentuk akar, caranya cukup kali bentuk sekawannya.
bentuk $ \sqrt{a} + \sqrt{b} $ sekawannya adalah $ \sqrt{a} - \sqrt{b} $
Perkaliannya : $ (\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) = a - b $
$\clubsuit $ Pembahasan
*). Kita Rasionakan masing-masing :
$ \begin{align} \frac{1}{1+\sqrt{2}} & = \frac{1}{1+\sqrt{2}} \times \frac{1- \sqrt{2}}{1-\sqrt{2}} = \frac{1- \sqrt{2}}{1-2} = -1+ \sqrt{2} \\ \frac{1}{\sqrt{2}+\sqrt{3}} & = \frac{1}{\sqrt{2}+\sqrt{3}} \times \frac{\sqrt{2}-\sqrt{3}}{\sqrt{2}-\sqrt{3}} = \frac{\sqrt{2}-\sqrt{3}}{2-3} = -\sqrt{2}+\sqrt{3} \\ \frac{1}{\sqrt{3}+\sqrt{4}} & = \frac{1}{\sqrt{3}+\sqrt{4}} \times \frac{\sqrt{3}-\sqrt{4}}{\sqrt{3}-\sqrt{4}} = \frac{\sqrt{3}-\sqrt{4}}{3-4} = -\sqrt{3}+\sqrt{4} \\ ........... & \\ \frac{1}{\sqrt{62}+\sqrt{63}} & = \frac{1}{\sqrt{62}+\sqrt{63}} \times \frac{\sqrt{62}-\sqrt{63}}{\sqrt{62}-\sqrt{63}} = \frac{\sqrt{62}-\sqrt{63}}{62-63} = -\sqrt{62}+\sqrt{63} \\ \frac{1}{\sqrt{63}+\sqrt{64}} & = \frac{1}{\sqrt{63}+\sqrt{64}} \times \frac{\sqrt{63}-\sqrt{64}}{\sqrt{63}-\sqrt{64}} = \frac{\sqrt{63}-\sqrt{64}}{63-64} = -\sqrt{63}+\sqrt{64} \end{align} $
*). Menjumlahkan semuanya :
$ \begin{align} & \frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}} + ... + \frac{1}{\sqrt{63}+\sqrt{64}} \\ & = (-1+ \sqrt{2}) + (-\sqrt{2}+\sqrt{3}) + .... + (-\sqrt{62}+\sqrt{63}) + (-\sqrt{63}+\sqrt{64}) \\ & = -1+ \sqrt{2} -\sqrt{2}+\sqrt{3} -\sqrt{3}+\sqrt{4} + .... -\sqrt{62}+\sqrt{63} -\sqrt{63}+\sqrt{64} \\ & = -1+ \sqrt{64} = -1+ 8 = 7 \end{align} $
Jadi, hasilnya adalah $ 7 . \, \heartsuit $
*). Kita Rasionakan masing-masing :
$ \begin{align} \frac{1}{1+\sqrt{2}} & = \frac{1}{1+\sqrt{2}} \times \frac{1- \sqrt{2}}{1-\sqrt{2}} = \frac{1- \sqrt{2}}{1-2} = -1+ \sqrt{2} \\ \frac{1}{\sqrt{2}+\sqrt{3}} & = \frac{1}{\sqrt{2}+\sqrt{3}} \times \frac{\sqrt{2}-\sqrt{3}}{\sqrt{2}-\sqrt{3}} = \frac{\sqrt{2}-\sqrt{3}}{2-3} = -\sqrt{2}+\sqrt{3} \\ \frac{1}{\sqrt{3}+\sqrt{4}} & = \frac{1}{\sqrt{3}+\sqrt{4}} \times \frac{\sqrt{3}-\sqrt{4}}{\sqrt{3}-\sqrt{4}} = \frac{\sqrt{3}-\sqrt{4}}{3-4} = -\sqrt{3}+\sqrt{4} \\ ........... & \\ \frac{1}{\sqrt{62}+\sqrt{63}} & = \frac{1}{\sqrt{62}+\sqrt{63}} \times \frac{\sqrt{62}-\sqrt{63}}{\sqrt{62}-\sqrt{63}} = \frac{\sqrt{62}-\sqrt{63}}{62-63} = -\sqrt{62}+\sqrt{63} \\ \frac{1}{\sqrt{63}+\sqrt{64}} & = \frac{1}{\sqrt{63}+\sqrt{64}} \times \frac{\sqrt{63}-\sqrt{64}}{\sqrt{63}-\sqrt{64}} = \frac{\sqrt{63}-\sqrt{64}}{63-64} = -\sqrt{63}+\sqrt{64} \end{align} $
*). Menjumlahkan semuanya :
$ \begin{align} & \frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}} + ... + \frac{1}{\sqrt{63}+\sqrt{64}} \\ & = (-1+ \sqrt{2}) + (-\sqrt{2}+\sqrt{3}) + .... + (-\sqrt{62}+\sqrt{63}) + (-\sqrt{63}+\sqrt{64}) \\ & = -1+ \sqrt{2} -\sqrt{2}+\sqrt{3} -\sqrt{3}+\sqrt{4} + .... -\sqrt{62}+\sqrt{63} -\sqrt{63}+\sqrt{64} \\ & = -1+ \sqrt{64} = -1+ 8 = 7 \end{align} $
Jadi, hasilnya adalah $ 7 . \, \heartsuit $
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