Question

1/6+1/12+1/20+1/30+.......+1/9900=?

Answer 1

Answer:

The value is $\frac{49}{100}=0.49$

Step-by-step explanation:

$\frac{1}{6}=\frac{1}{2}-\frac{1}{3}\\\frac{1}{12}=\frac{1}{3}-\frac{1}{4}\\\frac{1}{20}=\frac{1}{4}-\frac{1}{5}\\\frac{1}{30}=\frac{1}{5}-\frac{1}{6}$

Let,

$\frac{1}{9900}=\frac{1}{x}-\frac{1}{x+1}\\\frac{1}{9900}=\frac{x+1-x}{x(x+1)}\\\frac{1}{9900}=\frac{1}{x^{2}+x}\\9900=x^{2} +x\\x^{2}+x-9900=0\\x^{2}+100x-99x-9900=0\\x(x+100)-99(x+100)=0\\(x+100)(x-99)=0$

∴x = -100 or 99

∵x>0, ∴x = 99

∴$\frac{1}{9900}=\frac{1}{99}-\frac{1}{100}$

∴$\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\frac{1}{30}+......+\frac{1}{9900}\\=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4} -\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+...... -\frac{1}{99}+\frac{1}{99}-\frac{1}{100}\\=\frac{1}{2}-\frac{1}{100}\\=\frac{50-1}{100}\\=\frac{49}{100}\\ =0.49$

Hence required value = 0.49

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