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Question 184905: I am having difficulty adding positive odd integers. Please exlain in the simplest form (I have to present this to my teacher) the following question:
1+3=4
1+3+5=9
1+3+5+7=16
1+3+5+7+9=25
what is the sum of 1+3+5+7+9+....+99? Why?
I know I have to use (2n-1) but dont know how. Please help!
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! 1+3=4
1+3+5=9
1+3+5+7=16
1+3+5+7+9=25
what is the sum of 1+3+5+7+9+....+99? Why?
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Comment: Every odd number can be written in the form 2n-1 where
n is an integer. So another way of writing your problems is as follows:
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(2*1-1) + (2*2-1) = 2(1+2)-2 = 6-2 = 4
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(2*1-1)+(2*2-1)+(2*3-1) = 2(1+2+3)-3 = 12-3 = 9
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(2*1-1)+(2*2-1)+(2*3-1)+(2*4-1) = 2(1+2+3+4)-4 = 20-4 = 16
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Are you seeing the pattern?
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Now the last line: what is the sum of 1+3+5+7+9+....+99? Why?
The last term is 2n-1 = 99; Solve for n to get n = 50
So the sum is 2(1+2+3+...+49+50)-50 = 2-50 = [50*51]-50
= 50*50 = 2500
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I hope that helps.
Cheers,
Stan H.
I am having difficulty adding positive odd integers. Please exlain in the simplest form (I have to present this to my teacher) the following question:
1+3=4
Ans: So, 1+3 = 4
--------------------
1+3+5=9
Ans: (1+5)+3 = 6+3 = 9
-------------------------
1+3+5+7=16
Ans: (1+7)+(3+5) = 8+8 = 16
--------------------------
1+3+5+7+9=25
Ans: (1+9) + (3+7) + 5 = 2*10 + 5 = 25
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what is the sum of 1+3+5+7+9+....+99? Why?
Ans: (1+99) + (3+97) + (5+95) + (7+93) + ....(49+51) = ?
Do you see the pattern?
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Another way to do it:
1+3=4
Ans: 1 is 2*1-1 ; 3 is 2*2-1
So 1+3 = 2*1-1 + 2*2-1 = (2*1+2*2)-1-1 = 2(1+2)-2 = 2*3-2= 6-2= 4
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1+3+5=9
1+3+5 = 2(1+2+3)-1-1-1 = 2(6)-3 = 9
------------------------
1+3+5+7=16
1+3+5+7 = 2(1+2+3+4)-4 = 20-4= 16
--------------------------
1+3+5+7+9=25
= 2(1+2+3+4+5)-5=2*15-5 = 30-5=25
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what is the sum of 1+3+5+7+9+....+99? Why?
2(1+2+3+...+50)-50= 2*(50/2)*51)-50 = 2550-50 = 2500
========================
Cheers,
Stan H.
I know I have to use (2n-1) but dont know how. Please help!
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