Question 23924: My kid came at me with this and it does not make sense to me??
1+2+3+4+5+...+M=820
and the problem says solve for M
If it were not for the three dots
Answer by longjonsilver(2297)   (Show Source):
You can put this solution on YOUR website! the 3 dots just mean there are more numbers between the 5 and the final number, M.
If the Q had said 1+2+3+4+5+M = 820, then M is found easily.
This is a question of Arithmetic Series/Progressions. There are standard formula to answer this sort of question, that your son/daughter should have done. Your child really needs to understand this topic, since this question is quite tricky and long winded.
Anyway, I shall show you how to solve this "from first principles". This way, the question is answered completely, but is also very long, i'm afraid.
1 + 2 + 3 + 4 + ... + m-3 + m-2 + m-1 + m is a more mathematically robust "sum". Look at the last 4 "numbers" and see that they make sense. Whatever m is, say 100, then the number before it is 99..ie 100-1. The number before that is 98 ie 100-2 etc.
Anyway,
1 + 2 + 3 + 4 + ... + m-3 + m-2 + m-1 + m = 820
also
m + m-1 + m-2 + m-3 + ... + 4 + 3 + 2 + 1 = 820
Now, this second sum is the same as the first...just written back to front
Now add these 2 sums together... we get
m+1 + m+1 + m+1 + m+1 + ... + m+1 + m+1 + m+1 + m+1 = 1640
The issue is...how many m+1 terms are there? if there were say 10, then we have 10(m+1)=1640, from which we can solve for m. We do not know how many terms there are, so we call this n and we have n(m+1)=1640. One equation with 2 unknowns cannot be solved
So, we need another piece of information. This comes from looking at the terms themselves. In this example...a little thing you need to spot/figure out:
1+2+3...3 terms
1+2+3+4...4 terms
1+2+3+4+5...5 terms
so
1+2+3+4+...+m is m terms! ie our "n" in the previous equation is in fact equal to m!, so this tells us that the last number in this sequence is also how many terms we have in the sequence. We can use this and sub it into the first equation, namely n(m+1)=1640 which becomes m(m+1)=1640
This is now a quadratic: 
--> 
Now, either factorise this (i wouldn't bother...take you a while to figure out if it did factorise easily). Instead use the quadratic formula. UIsing this, gives m = 40 as one of the answers.
So, we have 1+2+3+4+...+38+39+40 = 840.
And doing that by hand on a calculator would take a lot longer than doing it the Arithmetic Series way :-)
Hope this helps
jon.
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