Why is 1 + 2 +...+ 2005 = 1/2 x 2005 x 2006?

Answer 1

because 2005+1=2006, 2004+2=2006, 2003+3=2006... 1002+1004=2006 - the total of 1002 sums, each equal 2006, plus 1003 (=2006/2), left without a pair.
1002*2006+2006/2=
=2006*(1002+1/2)=
=2006*1002.5=2006*2005/2

Answer 2

We have:
1 + 2 + 3 + ... + 2003 + 2004 + 2005

It is the same as
= 2005 + 2004 + 2003 + ... + 3 + 2 + 1

If we add them, then we have
2006 + 2006 + 2006 + ... + 2006 + 2006 + 2006

or
= 2005 x 2006

Since above we added the sum to itself, it meansthat 2005 x 2006 is twice the sum we are looking for. therefore to get the original sum, we multiply it by 1/2
1 + 2 + 3 + ... + 2005 = 1/2 x 2005 x 2006

QED
^_^

Answer 3

Try looking at it like this:

x = 1 + 2 + ... + 2004 + 2005

x = 2005 + 2004 + ... + 2 + 1

Then add these up in columns.

2x = 2006 + 2006 + ... + 2006 + 2006
Now there are 2005 terms on the right hand side, each of which is 2006, so this amounts to:

2x = 2005 * 2006
or
x = 1/2 * 2005 * 2006

Answer 4

this is becoz in sequence and series for Arithmetic Progression there is a formula,

Sn = n/2 (a + l)

where n= number of terms
a= the first term series
l= the last term of the series

so for the series u gave that is 1 + 2 + 3 +....+ 2005

n = 2005
a = 1
l = 2005

so u'll get,

Sn = 2005/2 ( 1 + 2005)
= 2005/2 (2006)
= 1/2 x 2005 x 2006

hope i helped. Goood Luck !!!

Answer 5

Picture a rectangle of blocks, 2005 blocks high and 2006 blocks wide. Now, take a diagonal slice through the rectangle by removing blocks. Remove 2005 from the first column, 2004 from the second, 2003 from the third, etc. until you remove 0 from the two thousand sixth. You have columns of 0, 1, 2, 3, etc. up to 2005. You got this by removing half of the 2005x2006 area.

Answer 6

Ok, so every1 has the answer...those using formulas to solve are out 'cos you are using rote and not logic to solve.
Every low number (eg. 1) has equivalent high number (eg. 2005) which adds up to 2006. So when you go from 1, you will stop at 1002 with 1002 pairs and only 1003 is the unpaired singleton.
So you have 1002.5 pairs of 2006 and since 1/2 is 1/2 of a pair, you have 1002.5x2=2005 of 2006
i.e. 1/2x2005x2006.
Most have solved it by forumla rote rather than logic.
Sorry for being a busybody but "Nobody" has the best and simplest solution. :) (pardon the pun)

Answer 7

1 + 2005 = 2006
2 + 2004 = 2006
3 + 2003 = 2006
....
Here is how to work it

(2005/2) * 2006 = 1002.5 * 2006 = 2011015

Answer 8

Hi. The sum of numbers 1+2+3+4=10, (4*5)/2 equals 10. The same goes for any sequence length.

Answer 9

X X X X X X
X X X X X O
X X X X O O
X X X O O O
X X O O O O
X O O O O O
O O O O O O

So 2 times Sum(1 through 6) = 6 x 7. You can try this with triangles of any size. So this suggests Sum (1 through n) = (1/2) * n * (n+1).

This can be proved by induction. But the easiest way to intuitively see it is through something like my array of Xs and Os above.

Answer 10

This works for the sum of the sequence of any amount of numbers you want to use.