1. ## Arithmetic Series

Consider the Arithmetic progression with first term 1, common difference 3 and last term 100. If A is any set of 19 distinct numbers from the arithmetic progression, show that there must always be two numbers whose sum is 104.

Thank you!

2. Originally Posted by acc100jt
Consider the Arithmetic progression with first term 1, common difference 3 and last term 100. If A is any set of 19 distinct numbers from the arithmetic progression, show that there must always be two numbers whose sum is 104.

Thank you!
The series is $\displaystyle a_n=2n-2,\ n=1, .. , 34$. That the $\displaystyle n$ and $\displaystyle$$m$ th terms sum to $\displaystyle 104$ means:

$\displaystyle (3n-2)+(3m-2)=104$

or:

$\displaystyle n+m=36$

Now both of $\displaystyle n=17$ and $\displaystyle m=19$ are in any set of $\displaystyle 19$ consecutive integers between $\displaystyle 1$ and $\displaystyle 34$ inclusive, etc.. .

CB

3. What happen if i pick 19 integers between 1 and 34 inclusive, but may not be consectutive, then n=17 and m=19 may not be in the set...hmm

4. Originally Posted by acc100jt
What happen if i pick 19 integers between 1 and 34 inclusive, but may not be consectutive, then n=17 and m=19 may not be in the set...hmm
Oppss.. misread the question, will get back to you on that

CB