Question 1177060
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Obviously, from the form of the expression, the statement of the problem is supposed to say that the closed form of the sum is for n >=1 -- not for n <= 1.<br>
There are only two unknowns in the problem -- a and b.  So two equations in a and b should be enough to solve the problem.<br>
n=1....<br>
{{{12(2(1^2)) = 12(2) = 24 = (1)(2)(3)(a(1)+b) = 6(a+b)}}} --> {{{a+b = 4}}}<br>
n=2....<br>
{{{12(2(1^2)+3(2^2)) = 12(2+12) = 168 = (2)(3)(4)(a(2)+b) = 24(2a+b)}}}  -->  {{{2a+b = 7}}}<br>
a+b=4; 2a+b=7 --> a=3, b=1.<br>
ANSWER: an+b = 3n+1<br>
CHECK for n=3....<br>
12(2(1^)+3(2^2)+4(3^2) = 12(2+12+36) = 600
(3)(4)(5)(3(3)+1) = 600<br>