ap arithmatic progression

[the arrivals]

the smallest positive integer x for which the sum x+2x+3X+4x+5x+...........100x is a perfect square is ??
suppose we ve the system of equation
a+b+c=10
a+c=3
a^2+b^2=65
can you guess the value of c??
( a little advice for solving this problems.u can suppose a,b,c are the roots of a 3degree polynomial equation.sometimes u can also consider that they are the sides of a trianle and then u can play with a,b,c like as u play with a triangle.u can derive new formula using cosine or sine method
but beleive me this piece isnt that much cruel

[Masum]

$1.x+2x+....+100x=25.2.3.17,x_{min}=2.3.17$
$2.b=7,a=\pm 4,$that's enough