[Ghostavo]

A few days ago I was given this problem and I am unable to find a solution. You are given the following sum:

1¹+2²+3³+(...)+2003²°°³+2004^2004+2005^2005

To which you must find the unit's digit, all of this with just a piece of paper, a pencil and your head.



Help?

[adwight]

You do the thing where you take the first 5 numbers on top, then the next 5 and add them I believe so it's something like

1^1, 2^2, 3^3
2005^2005

Im sure someone else knows how to do it but thats the way i would think it would be done.(if you understand what I am saying)

[Corsair]

So basically...

2005
r^r
r=1

I probably should know how to do this.

[Mr. Vega]

Corsair, how big is that piece of paper?

[ShadowWolf_IH]

ok so if you were paddling your canoe down mainstreet and all four tires fell off how many pancakes does it take to make a doghouse?

[Solatar]

Pi minus a Bm7.

[Corsair]

Originally posted by Mr. Vega
Corsair, how big is that piece of paper?

Small handwriting.

Hey Solatar, what's a Bm7?

[an0n]

Correct me if I'm wrong, but isn't this just a case of using that stupid geometric progression formula thingy?

[Bobboau]

that problem made my TI89 cry

[Solatar]

Originally posted by Corsair


Hey Solatar, what's a Bm7?

B minor 7th, it's a music chord.

[Ghostavo]

adwight, could you be more specific? I can't understand what you mean.

Corsair, this was to be done in less than 2 hours (this was part of a 4 exercise "test"). Yeah that's it, but after reaching there the whole thing goes to hell, if I had about 1 or 2 days I'd probably do it the hard way (calculate it all) but this was not the case as I said before.

an0n, explain?

[an0n]

If I could 'explain', I'd've tried to apply my theory to the problem by now.

All I've got is a vague memory of some quick way to figure out geometric progressions (assuming that's what this is).

[Ghostavo]

But this is not geometric, it's exponencial...

[Joey_21]

I came up with this right after Ghostavo left #nodewar:

< Joey_21 > 1, 4, 7, 6, 5, 6, 3, 6, 9, 0, 1, 6, 3, 6, 5, 6, 7, 4, 9, 0
< Joey_21 > next digits: 1, 4, 7, 6, 5, 6, 3, 6, 9, 0...
< Joey_21 > 1^1 ends in 1, 2^2 ends in 4, 3^3 ends in 7, 4^4 ends in 6.... 20^20 ends in a 0, there the pattern starts repeating

< Joey_21 > 2000^2000 ends in 0, 2001^2001 ends in 1, 2002^2002 ends in 4, 2003^2003 ends in 7, 2004^2004 ends in 6, 2005^2005 ends in 5
< Joey_21 > there we see the sequence 0, 1, 4, 7, 6, 5 again

I don't have the time to continue it from here because I have to go to class in a few minutes. Good luck!

[Corsair]

Originally posted by Solatar


B minor 7th, it's a music chord.

D'oh. I knew that. I thought you meant something in math, not music.

an0n, like Ghostavo said, it isn't geometric so you can't use those formulas because they don't describe the increasing exponent.