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#1 2006-10-31 06:29:09

Patrick
Real Member
Registered: 2006-02-24
Posts: 1,005

Sum of the 12 first odd numbers

This hasn't really been discussed in my class, and I don't think it will anytime soon. Say an amount of girls, n, went to the cinema(or took an air-plane) and had to sit in chairs numbered 1, 3, 5 ... The number of girls isn't known, but the sum of their chair numbers is 144. If I knew the amount of girls, I'd find the sum using:

Using mathematica I found that the problem could be solved with the following equation:

Where S is the sum of all odd numbers up to the nth odd number. Simplifying, this gives me:

Obviously, by now I would be able to solve my original problem with ease:

So, here's my question:
How do I go from

to

?

A general approach would be highly appreciated! Also if I've understood something wrong, please correct me

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#2 2006-10-31 06:47:22

luca-deltodesco
Member
Registered: 2006-05-05
Posts: 1,470

Re: Sum of the 12 first odd numbers

The Beginning Of All Things To End.
The End Of All Things To Come.

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#3 2006-10-31 07:06:01

Patrick
Real Member
Registered: 2006-02-24
Posts: 1,005

Re: Sum of the 12 first odd numbers

Thanks alot luca-deltodesco!
I'm not sure if I understand why

edit: Aha! Arithmetic series

Last edited by Patrick (2006-10-31 07:16:46)

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#4 2006-10-31 07:16:04

luca-deltodesco
Member
Registered: 2006-05-05
Posts: 1,470

Re: Sum of the 12 first odd numbers

if you want a proof for the 0.5n(n+1) thing. then here it is

what i want to prove is that

S(1) by expanding the sum, is ofcourse 1.

so the formula holds for n = 1

assuming that the formula is true for any n, i need to prove that it is true for any n+1

which is the above formula. So ive now shown that the formula holds for n = 1, and for any (n+1), given that it works for S(n)

by process of induction, this means that any integer n, n>=1, the sum of all integers 1 to n, is equal to S(n), since S(1) works, with the last set of formulae, S(2) must also work, which means S(3) must then also work too, and so on

Last edited by luca-deltodesco (2006-10-31 07:19:07)

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#5 2006-10-31 07:18:32

Patrick
Real Member
Registered: 2006-02-24
Posts: 1,005

Re: Sum of the 12 first odd numbers

okay, that made it a bit clearer than wikipedia... Thank you very much!

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