fill with different natural numbers: (1/(19))=(1/(( )))+(1/(( )))+(1/(( )))+(1/(( )))


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Question Number 194144 by mr W last updated on 28/Jun/23

$f i l l w i t h d i f f e r e n t n a t u r a l n u m b e r s :$ $\frac{1}{19} = \frac{1}{()} + \frac{1}{()} + \frac{1}{()} + \frac{1}{()}$
	Answered by York12 last updated on 28/Jun/23
	$egyptian fractions (1/k)=(1/(k+1))+(1/(k(k+1)))$
	$e g y p t i a n f r a c t i o n s$ $\frac{1}{k} = \frac{1}{k + 1} + \frac{1}{k (k + 1)}$
	Answered by BaliramKumar last updated on 28/Jun/23

	$\frac{1}{19} = \frac{1}{(20)} + \frac{1}{(381)} + \frac{1}{(144781)} + \frac{1}{(20961393180)}$
	Commented by Frix last updated on 28/Jun/23

	$\frac{1}{21} + \frac{1}{381} + \frac{1}{420} + \frac{1}{144780}$
	Commented by BaliramKumar last updated on 28/Jun/23

	$Also$ $\frac{1}{19} = \frac{1}{19 \times 20} + \frac{1}{20 \times 21} + \frac{1}{21 \times 22} + \frac{1}{22}$ $\frac{1}{19} = \frac{1}{(22)} + \frac{1}{(380)} + \frac{1}{(420)} + \frac{1}{(462)}$ $\begin{matrix} \frac{1}{n} = \frac{1}{n (n + 1)} + \frac{1}{(n + 1) (n + 2)} + \frac{1}{(n + 2) (n + 3)} + \frac{1}{(n + 3)} . . . . . . \end{matrix}$
	Answered by witcher3 last updated on 28/Jun/23

	$\frac{1}{19} = \frac{1}{19} . 1$ $1 = \frac{10}{10} = \frac{5 + 2 + 2 + 1}{10} = \frac{1}{2} + \frac{1}{5} + \frac{1}{5} + \frac{1}{10}$
	Commented by BaliramKumar last updated on 28/Jun/23

	$Nice approach but \frac{1}{5} & \frac{1}{5} same number$ $\frac{1}{19} = \frac{1}{19} \times \frac{20}{20} = \frac{1}{19} (\frac{10 + 5 + 4 + 1}{20})$ $\frac{1}{19} = \frac{1}{19} (\frac{1}{2} + \frac{1}{4} + \frac{1}{5} + \frac{1}{20})$ $\frac{1}{19} = \frac{1}{2 \times 19} + \frac{1}{4 \times 19} + \frac{1}{5 \times 19} + \frac{1}{20 \times 19}$ $\frac{1}{19} = \frac{1}{38} + \frac{1}{76} + \frac{1}{95} + \frac{1}{380}$
	Commented by York12 last updated on 28/Jun/23
	$there are a lot actually as I have mentioned above it is an extension to the egyptian fractions which are still a mystrey till today$
	$t h e r e a r e a l o t a c t u a l l y a s I h a v e m e n t i o n e d$ $a b o v e i t i s a n e x t e n s i o n t o$ $t h e e g y p t i a n f r a c t i o n s w h i c h a r e s t i l l$ $a m y s t r e y t i l l t o d a y$
	Commented by MM42 last updated on 28/Jun/23

	$i f d_{i} ∣ m; 1 ⩽ i ⩽ k & Σ d_{i} = n$ $\Rightarrow \frac{Σ d_{i}}{m} = \frac{1}{m_{1}} + \frac{1}{m_{2}} + . . . + \frac{1}{m_{k}}; m_{i} = \frac{m}{d_{i}}$ $f o r e x a m p l e :$ $\frac{1}{24} = \frac{12 + 8 + 3 + 1}{24} = \frac{1}{12} + \frac{1}{3} + \frac{1}{8} + \frac{1}{24}$ $o r \frac{1}{24} = \frac{8 + 4 + 6 + 3 + 2 + 1}{24} = \frac{1}{3} + \frac{1}{6} + \frac{1}{4} + \frac{1}{8} + \frac{1}{24}$
	Answered by Spillover last updated on 01/Jul/23
	$By using Egptian fraction.we can express (1/(19)) as sum of unit fraction (1/(19))=(1/((3)))+(1/((37)))+(1/((228)))+(1/((342)))$
	$B y u s i n g E g p t i a n f r a c t i o n . w e c a n e x p r e s s$ $\frac{1}{19} a s s u m o f u n i t f r a c t i o n$ $\frac{1}{19} = \frac{1}{(3)} + \frac{1}{(37)} + \frac{1}{(228)} + \frac{1}{(342)}$
	Answered by Spillover last updated on 01/Jul/23

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