BdMO National 2012: Primary 6

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Zzzz
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BdMO National 2012: Primary 6

Unread post by Zzzz » Sun Feb 12, 2012 1:56 pm

Problem 6:
Consider the given diagram. There are three rectangles shown here. Their lengths are $3,\ 4$ and $5$ units respectively, widths respectively $2,\ 3$ and $4$ units. Each small grid represents a square $1$ unit long and $1$ unit wide. Use these diagrams to find out the sum of the consecutive numbers from $1$ to $500$. (If you use some direct formula for doing so, you must provide its proof)
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samiul_samin
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Re: BdMO National 2012: Primary 6

Unread post by samiul_samin » Wed Feb 21, 2018 12:01 am

I have got a nice solution just now.It is easy.
Solution
According to the picture,
(The area of the rectangles are shared by same number of light colored square and dark colored square and they are increasing one by one in every diagram.)

The area of rectangle $1=3×2=2(1+2)$
The area of rectangle $2=4×3=12=2(1+2+3)$
The area of rectangle $ 3=5×4=20=2(1+2+3+4)$
The area of rectangle $ 4=6×5=30=2(1+2+3+4+5)$
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The area of the rectangle $499=501×500=2(1+2+3+4+5+6+7+8+9+...+499+500)$

So,$1+2+3+4+5+6+7+8+9+..+499+500=\dfrac {1}{2}(500×501)=125250$
The desired answer is $\fbox {125250}$
No,need to prove any formula.It is really easy.

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