SOLUTION: Solve the equation2+5+8+11+.........+x=345

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Question 898615: Solve the equation2+5+8+11+.........+x=345
Answer by ewatrrr(24785) About Me  (Show Source):
You can put this solution on YOUR website!
 

Hi

Arithmetic : d = 3

S%5Bn%5D=+%28n%2F2%29%282%2Aa%5B1%5D%2B+%28n-1%29d%29+=+345

%28n%2F2%29%282%2A2%2B+3n+-+3%29+=+345

%28n%2F2%29%283n%2B1%29=+345

 3n^2 + n = 690

 3n^2 + n - 690 = 0

 


 
Solved by pluggable solver: SOLVE quadratic equation with variable


Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 3x%5E2%2B1x%2B-690+=+0) has the following solutons:

  

  x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

  

  For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

  

  First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%281%29%5E2-4%2A3%2A-690=8281.

  

  Discriminant d=8281 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-1%2B-sqrt%28+8281+%29%29%2F2%5Ca.

  

    x%5B1%5D+=+%28-%281%29%2Bsqrt%28+8281+%29%29%2F2%5C3+=+15

    x%5B2%5D+=+%28-%281%29-sqrt%28+8281+%29%29%2F2%5C3+=+-15.3333333333333

    

    Quadratic expression 3x%5E2%2B1x%2B-690 can be factored:

  3x%5E2%2B1x%2B-690+=+3%28x-15%29%2A%28x--15.3333333333333%29

  Again, the answer is: 15, -15.3333333333333.
Here's your graph:

graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B1%2Ax%2B-690+%29





n = 15 

a%5Bn%5D=a%5B1%5D%2B%28n-1%29%2Ad

 a%5B15%5D=+2+%2B%2814%29%2A3+=+44

   x = 44

check: 15(2 + 44)/2 = 345