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The ghost ship [The Problem of the Week] Sums of odd

Today, another of the old problems of Dodka:

We have commissioned let us add all the odd numbers from 1 to 101 inclusive. After a while We have completed the account, but then we have commissioned the sum of 1 to 201, and tired of adding, we want to find an easy formula to give us the result.

Find the formula, and as application calculates the sum of the odd numbers from 1 to 101, from 1 to 201 and from 1 to 343.

Where is the solution? Just past

illustration. Girard

wheel on your mouse;

the

find

without delay.

[This image is taken from an activity by George Hart at Albion College (Albion College ) in the summer of 2008. Shows the lobby of the university, which led Hart to various groups of people to assemble the geometric structures which were then hoisted and hung at different heights, as shown in the photo. In total there are nine shapes, even in the picture are only eight, and are positioned so that its sequence inspired by the flight of an asteroid or comet in space. Surprisingly, although the structures are perfectly regular, Hart has made an intricate shape that together with the chosen colors suggest the irregular shape and rugged incandescent rock entering the Earth's atmosphere. See note at end of this post for more details on George Hart]

Solution:

Just keep trying with the first sums of odd and realize what's happening:

1 = 1 = 1

2 1 + 3 = 4 = 2 2

1 + 3 + 5 = 9 = 3 2 ...

The sums of consecutive odd numbers always gives the square of a number, specifically the number being the odd arithmetic mean we used. If we add up all the odd numbers from 1 to 101 will leave the square of the number that is average between 1 and 101:

(1 + 101): 2 = 51, then

1 + 3 + 5 + 7 + ... + 99 + 101 = 51 = 2 2601

And in the same way:

1 + 3 + 5 + 7 + ... + 199 + 201 = 101 = 2 10201

1 + 3 + 5 + 7 + ... + 341 + 343 = 172 = 2 29584

Note: obviously, for the matenavegantes a little tanned, this problem is actually the sum arithmetic progressions, and you can use formulas to calculate these sums, but in this particular case, adding odd numbers, we found that the shortcut explained above and the formulas are not necessary, just a little care and ingenuity.

Note: George Hart page is one of the latest discoveries in our journey of matenavegación. We recommend that all school officials Ships that visit, because it is full of exciting activities. One of them, namely that Hart held at the University of Seville in October 2008, we have repeated in our own boat, and this link you can see the results.