Question 336261
arithmetic series:  {{{a[1] + (a[1]+diff)+(a[1]+2*diff)}}} +...+ {{{(a[1]+(n-1)*diff)=sum((a[1]+(i-1)*diff)))}}} summed from i=1 to n
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Given {{{a[1]=-1}}}, diff=5 and sum=1356
{{{sum((a[1]+(i-1)*diff))=sum((-1+(i-1)*5))=1356}}} 
{{{sum((5i-6))=1356}}}
{{{5*sum(i)-n*6=1356}}}

{{{5*n*(n+1)/2-6*n=1356}}}
{{{5*n*(n-1)-12*n=2712}}}  (multiply both sides by 2)
{{{5n^2+5*n-12*n=2712}}}
{{{5*n^2-7*n-2712=0}}}
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using quadratic solution
with a=5, b=-7, c=-2712
{{{n=(7-Sqrt(7^2-4*(5)*(-2712)))/(2*5)=(7-233)/10=-22.6}}}
or
{{{n=(7+Sqrt(7^2-4*(5)*(-2712)))/(2*5)=(7+233)/10=24}}}
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since we need the solution to be positive, the first solution is not needed
and n=24