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 2 years ago
What is the sum of all values of N, 1≤N≤999, such that 1+2+3+…+N is a perfect square?
 2 years ago
What is the sum of all values of N, 1≤N≤999, such that 1+2+3+…+N is a perfect square?

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Traxter
 2 years ago
Best ResponseYou've already chosen the best response.0The sum n n numbers from 1 to n is given by: \[\frac{n(n+1)}{2}\] Are you asking for a value of N such that the sum is a perfect square, or for the sum of numbers from 1 to 999?

irudayadhason
 2 years ago
Best ResponseYou've already chosen the best response.0there are many numbers for N i am asking their sums

Jonask
 2 years ago
Best ResponseYou've already chosen the best response.0@Traxter gave you the sums of natural numbers

Traxter
 2 years ago
Best ResponseYou've already chosen the best response.0It might be helpful to note that the sum of n odd numbers is a perfect square: \[1+3+5+...+(2n1)=n^2\] So we want the sum of all the odd numbers betwee 1 and 999. So take (2n1)=999, so that 2n=1000 and n=500. So The sum of all the odd numbers between 1 and 999 is \(500^2\)=250,000.

Traxter
 2 years ago
Best ResponseYou've already chosen the best response.0I think that's what you're looking for @irudayadhason.

irudayadhason
 2 years ago
Best ResponseYou've already chosen the best response.0the answer should be an integer between 0999

Traxter
 2 years ago
Best ResponseYou've already chosen the best response.0The question really doesn't make much sense, do you think you could rephrase it please?

irudayadhason
 2 years ago
Best ResponseYou've already chosen the best response.0no there are no mistakes in this question

Jonask
 2 years ago
Best ResponseYou've already chosen the best response.0i think he is looking for values of N but all the answers added up @irudayadhason

mukushla
 2 years ago
Best ResponseYou've already chosen the best response.2so we want to solve\[1+2+3+...+n=m^2\]\[n(n+1)=2m^2\]

irudayadhason
 2 years ago
Best ResponseYou've already chosen the best response.0no i am not looking for the values of N i am just asking the sum of all values of N but it should be between 0999

asnaseer
 2 years ago
Best ResponseYou've already chosen the best response.0Maybe this will help explain it to others better: lets say the only solutions to this were n=1 and n=8, then your answer would be 1 + 8 = 9. so, from all the valid solutions to this (\(n_1,n_2,n_3,...\)), the answer we want is:\[n_1+n_2+n_3+...\]

irudayadhason
 2 years ago
Best ResponseYou've already chosen the best response.0sorry 9 is a wrong answer try to use this key technique "Pell's Equation."

asnaseer
 2 years ago
Best ResponseYou've already chosen the best response.0I didn't say 9 way the answer  I was using it as an example to explain the question. :)

asnaseer
 2 years ago
Best ResponseYou've already chosen the best response.0the aim of this site is not "just to hand out answers". it is to teach and learn.

mukushla
 2 years ago
Best ResponseYou've already chosen the best response.2\[n(n+1)=2m^2\]\[4n(n+1)=8m^2\]\[(2n+1)^2=8m^2+1\]letting\[x=2n+1\]\[y=2m\]will gives\[x^22y^2=1\]

irudayadhason
 2 years ago
Best ResponseYou've already chosen the best response.0you mean the answer is ...

irudayadhason
 2 years ago
Best ResponseYou've already chosen the best response.0Traxter please try to solve

mukushla
 2 years ago
Best ResponseYou've already chosen the best response.2later equation solved by Pell see here http://en.wikipedia.org/wiki/Square_triangular_number

mukushla
 2 years ago
Best ResponseYou've already chosen the best response.2answer will be\[x=P_{2k}+P_{2k1}\]\[y=P_{2k}\]\(P_k\) is kth Pell number see here http://en.wikipedia.org/wiki/Pell_number

irudayadhason
 2 years ago
Best ResponseYou've already chosen the best response.0i need an answer between 0999
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