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April 5th, 2014, 09:27 AM   #1
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Question How to solve this consecutive numbers addition?

Hello everyone!

Please teach me how to solve this question:

Find the value of: 7+9+11+13+15+17+……+55+57+59+61

Ans: 952

and this question:

Find the value of: 10+20+3+4+5+……+6+7+8+9+10

Thanks!
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April 5th, 2014, 10:07 AM   #2
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Math Focus: Calculus/ODEs
For the first one, we could use the formula:

$\displaystyle \sum_{k=1}^n(2k-1)=n^2$

And so your sum $S$ is:

$\displaystyle S=\sum_{k=4}^{31}(2k-1)=\sum_{k=1}^{31}(2k-1)-\sum_{k=1}^{3}(2k-1)=31^2-3^2=(31+3)(31-3)=28\cdot34=952$
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April 5th, 2014, 10:26 AM   #3
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Hello MarkFL,

Thanks for sharing!

Can you share me the Primary Schools method too?

Thanks,
XP
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April 5th, 2014, 10:39 AM   #4
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Well, we could state:

$\displaystyle S=7+9+11+\cdots+57+59+61$

$\displaystyle S=61+59+57+\cdots+11+9+7$

Now, if we add each column, we find:

$\displaystyle 2S=68+68+68+\cdots+68+68+68$

Seeing that we have 28 68's on the right side, we may then write:

$\displaystyle 2S=28\cdot68$

Divide both sides by 2:

$\displaystyle S=28\cdot34=952$
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April 5th, 2014, 11:32 AM   #5
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Hi MarkFL,

The 2nd method looks easier but I don't understand.

1. What is "S"?
2. What is "2S"?
3. How did you know there are 28 68's on the right without calculating?
4. Final step, S= 28.34, how did you calculate into 952?
5. Add what column?

Do you still have any other easier and simpler steps?

Regards,
Mai
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April 5th, 2014, 11:43 AM   #6
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$S$ just stands for the value of the sum, and so $2S$ is twice the value of the sum.

$7=2\cdot4-1$ and $61=2\cdot31-1$, and so there are $31-4+1=28$ addends in the sum.

$\displaystyle 28\cdot34=4\cdot14\cdot17=4(15-1)(15+2)=4(225+15-2)=4(238)=4(240-2)=960-8=952$

If you look at the two expressions I wrote, then look from top to bottom at each term that lines up:

$\displaystyle S+S=(7+61)+(9+59)+(11+57)+\cdots+(57+11)+(59+9)+(6 1+7)$

$\displaystyle 2S=68+68+68+\cdots+68+68+68$

Last edited by MarkFL; April 5th, 2014 at 11:47 AM.
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April 5th, 2014, 09:15 PM   #7
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Hello MarkFL,

Then what about the dot (.) stand for?

Quote:
$\displaystyle 7 = 2.4-1$
$\displaystyle 2.4$ is a decimal or... ?

Best regards,
Mai
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April 5th, 2014, 10:01 PM   #8
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That dot is above where a decimal point would go and is a symbol for multiplication.
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April 5th, 2014, 10:55 PM   #9
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Hi MarkFL,

Do you have any other easier methods? Your 3rd method is more confusing.

Quote:
$\displaystyle 31−4+1=28$
Where did you get the 4 and 1 from?

Quote:
$\displaystyle 28⋅34$
Where did you get the 34 from?

By the way, the 2nd method I understood a little bit, I understood how you got 68 but not understand how you got 28 68's.

Quote:
$\displaystyle 2S=28⋅68$
Kind regards,
Mai

Last edited by XPMai; April 5th, 2014 at 11:04 PM.
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April 5th, 2014, 11:20 PM   #10
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Well, you could manually add the terms.

The 4 and 31 come from:

$\displaystyle 7=2\cdot4-1$

$\displaystyle 61=2\cdot31-1$

The 1 comes from the fact that the number of terms is the difference in indices plus 1.For example, consider if you have 10 objects labeled 1 through 10.

(10 - 1) + 1 = 10
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