SOLUTION: 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

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.

RELATED QUESTIONS
I need to find the intercepts, graph and check for symmetry. y=-5x, could someone show me (answered by doukungfoo)

m/m^2-m-12 + 5/m+3 This is what i came up with but i'm not exactly sure how i... (answered by stanbon)

My 8th grade son has the following problem: -5(x + 2) = -5x - 10 He is telling me... (answered by Edwin McCravy)

3*4^2-2^3-3^2=23 We were asked to put grouping symbols in this problem to make it... (answered by macston)

Dear math teacher, I really need help with the following word problem: There are... (answered by edjones)

May I get help with this problem. It just says to simplify. The answer I came up with... (answered by stanbon)

Hi, So my friends have been going crazy over a problem. I'm not sure of the exact... (answered by MathLover1)

Can you please help me understand this problem? Evaluate: xy^-2/3 for x=2 y=8... (answered by Alan3354)

Does this make sense for answers? I'm trying but I just want to double check... For each (answered by jim_thompson5910,ewatrrr)