anonymous
  • anonymous
Medal & Fan. The sum of the squares of 3 consecutive positive integers is 116. What are the numbers? Which of the following equations is used in the process of solving this problem? 3n^2 + 5 = 116 3n^2 + 3n + 3 = 116 3n^2 + 6n + 5 = 116
Mathematics
schrodinger
  • schrodinger
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anonymous
  • anonymous
freckles
  • freckles
an and example of 3 consecutive positive integers is: 1,2,3 or 3,4,5 ... or (n-1),n,(n+1) we don't know what they are so let's go with the (n-1),n,(n+1) being the 3 consecutive positive integers so you have the sum of the squares of them is 116 that is (n-1)^2+n^2+(n+1)^2=116 play with the left hand side
hybrik
  • hybrik
Freckles can do this one, I have to do something at the moment, be right back.

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anonymous
  • anonymous
2n+2+n+2n+2=116?
anonymous
  • anonymous
probly horribly wrong
freckles
  • freckles
(n-1)^2 = (n-1)(n-1) do you know how to expand this?
anonymous
  • anonymous
foil maybe?
freckles
  • freckles
yeah or it is just really distributing like n(n-1)-1(n-1)
anonymous
  • anonymous
n^ 2- 2n + 1
freckles
  • freckles
right and we can also expand (n+1)^2 in a similar way
freckles
  • freckles
which should be n^2+2n+1 for (n+1)^2 so you have this now: \[(n-1)^2+n^2+(n+1)^2 =116 \\ (n^2-2n+1)+n^2+(n^2+2n+1)=116 \]
freckles
  • freckles
combine like terms on the left hand side
anonymous
  • anonymous
\[-2n ^{2}+1+n^2 +2n^2+1=116\]
anonymous
  • anonymous
not done yet
freckles
  • freckles
don't go any further with what you have just wrote
freckles
  • freckles
in the equation I wrote how many n^2 's do you see?
anonymous
  • anonymous
3
freckles
  • freckles
n^2+n^2+n^2 is 3n^2
freckles
  • freckles
now let's look at the n's you have -2n+2n which equals ?
anonymous
  • anonymous
n
freckles
  • freckles
if someone gave you 3 apples and your three all 3 apples away how many apples do you have left?
anonymous
  • anonymous
0
freckles
  • freckles
if someone gave you 3 apples and your threw all 3 apples away how many apples do you have left?* yes 0
freckles
  • freckles
so -2n+2n is ?
anonymous
  • anonymous
0
freckles
  • freckles
0 also replace 2n with 5 you have -5+5 and we know that is 0 you can also write this -2n+2n both terms have a common factor n factor the n out n(-2+2) but you should know -2+2=0 so you have n(0) but 0 times anything will result in 0 so n(0)=0 so -2n+2n=0
freckles
  • freckles
the last thing to add are the constant terms (the terms with out any variable)
freckles
  • freckles
can you show me what you think you have?
anonymous
  • anonymous
the whole thing?
freckles
  • freckles
sure why not you probably want me to check the whole thing right?
anonymous
  • anonymous
3n^2+2=0?
freckles
  • freckles
yep but that doesn't match any of your answers I guess they chose their 3 consecutive numbers differently maybe they chose n,n+1,n+2 which also works \[n^2+(n+1)^2+(n+2)^2=116 \\ n^2+n^2+2n+1+n^2+4n+4=116 \\ 3n^2+6n+5=116 \]
freckles
  • freckles
you can choose the three consecutive integers in many ways
freckles
  • freckles
that is solving either one of the equations will result in the same 3 consecutive integers
anonymous
  • anonymous
thanks again
freckles
  • freckles
though I like my equation better because it seems simpler to solve :p

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