odd problem

[Ryan Rigdon]

I am asking help to explain this to my daughter who is in 6th grade. my idea was to do this

1+3+5+7+9+11+13 and so on to get the first 100 odd numbers. I came up with 10000. There is anyway I can show her on how to do this geometrically. Thank

you for your time and responses.

 

[Ishuda]

Not geometrically but how about arithmetically:

Conceptually write your numbers as
s = 1 + 3 + 5 + .... + 199
where 199 is the 100th odd number
Now rewrite it as
s = 199 + 198 + 197 + .... + 1

so add the two together to get double the amount
2 s = 200 + 200 + 200 + ... + 200
where the 200 occurs 199 time. Thus
2 s = 200 * 199
or
s = 19 900

maybe someone else will know how to do it geometrically or maybe you can turn this into a geometric show and tell [go 1 step, then go 3 steps, ... then go 199 steps. how many steps did you take? you could also have first gone 199 steps, then 198 steps, ...]

 

[wjm11]

Not geometrically but how about arithmetically:

Conceptually write your numbers as
s = 1 + 3 + 5 + .... + 199
where 199 is the 100th odd number
Now rewrite it as
s = 199 + 198 + 197 + .... + 1

so add the two together to get double the amount
2 s = 200 + 200 + 200 + ... + 200
where the 200 occurs 199 time. Thus
2 s = 200 * 199
or
s = 19 900

maybe someone else will know how to do it geometrically or maybe you can turn this into a geometric show and tell [go 1 step, then go 3 steps, ... then go 199 steps. how many steps did you take? you could also have first gone 199 steps, then 198 steps, ...]

The above steps demonstrate a nice method for finding the sum; however, there are a couple of typos to be aware of: "198" should not be in the second series, and the number of times that "200" occurs is not "199." Rather it is 100 occurrences. So, s = (200)(100)/2 = 10,000 (as the OP suggested). I do not yet recognize the "geometric" approach requested.

 

[Quaid]

Where are the dots?!

The missing dots form a square array, five by five. (Note the hint.)

5×5 = 25

The sum of the first five odd numbers is 25.

We could expand the diagram to the first nine odd numbers; those dots would form a nine by nine array.

9×9 = 81

The sum of the first nine odd numbers is 81.

The sum of the first 25 odd numbers is 625 (25×25).

The sum of the first 66 odd numbers is 4356 (66×66).

And so on.

 

[Ishuda]

The above steps demonstrate a nice method for finding the sum; however, there are a couple of typos to be aware of: "198" should not be in the second series, and the number of times that "200" occurs is not "199." Rather it is 100 occurrences. So, s = (200)(100)/2 = 10,000 (as the OP suggested). I do not yet recognize the "geometric" approach requested.

Oh, I though 198 was odd like me - yes of course you're correct about that and somehow I thought when one said the number of numbers was 100, they really meant 199. Oh well live and learn. I'll stop there for this problem, no telling what trouble I'll get into if I continue.