Question 1209781
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Answer: <font color=red>13717</font>



Work Shown
{{{sum((5n + 2 - 4n + n^2 + 17),n=1,33)}}}


= {{{sum((n^2 + n + 19),n=1,33)}}}


= {{{(sum(n^2,n=1,33))+(sum(n^"",n=1,33))+(sum(19^"",n=1,33))}}}


= {{{(1/6)*33*(33+1)*(2*33+1)+(1/2)*33*(33+1)+33*19}}} See formulas below


= {{{13717}}}


Therefore,
{{{sum((5n + 2 - 4n + n^2 + 17),n=1,33) = 13717}}}


Verification using <a href="https://www.wolframalpha.com/input?i=sum+5n+%2B+2+-+4n+%2B+n%5E2+%2B+17+from+n+%3D+1+to+n+%3D+33">WolframAlpha</a>


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Formulas Used
{{{sum(k^2,k=1,n) = matrix(1,3,1^2+2^2+3^2,"+...+",n^2) = (1/6)n(n+1)(2n+1)}}}


{{{sum(k^"",k=1,n) = matrix(1,3,1+2+3,"+...+",n) = (1/2)n(n+1)}}}


{{{drawing(220,150,-5,5,-5,5,
arc(-0.736,0.126,0.868,0.868,90,180),line(-0.736,-0.308,0.566,-0.308),arc(0.566,-0.742,0.868,0.868,270,360),arc(1.434,-0.742,0.868,0.868,180,270),line(1.434,-0.308,2.736,-0.308),arc(2.736,0.126,0.868,0.868,0,90),
locate(-4,3,sum(c^"",k=1,n) = matrix(1,3,c+c+c,"+...+",c) = cn),
locate(-1,-1,matrix(1,4,"n","copies","of","c")),
locate(-1,-2,matrix(1,2,"being","added"))
)
}}}
where c is a constant
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