Question 842180
The first term of the SEQUENCE is 19.
The difference between terms is NEGATIVE 3.
The n term of the SEQUENCE is {{{19+(n-1)*(-3)}}}.


The formula for Arithmetic SERIES is {{{(n/2)(a[1]+a[n])}}}.

You want this equation:  {{{highlight_green((n/2)(19+19+(n-1)*(-3))=44)}}}.


SOLUTION STEPS
{{{(n/2)(38+(-3n+3))=44}}}
{{{n(41-3n)=88}}}
{{{41n-3n^2=88}}}
{{{3n^2-41n+88=0}}}
{{{Discriminant = 625}}} and {{{sqrt(625)=25}}}; this is done for using general solution to a quadratic equation and avoid trying to factor the quadratic expression.
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{{{n=(41+- 25)/(6)}}} without showing all the steps for this formula.
The MINUS form will give a bad value, a mixed number while the PLUS form gives {{{highlight(highlight(n=11))}}}.


What you might try, to be sure, find the sum of this:
{{{19+16+13+10+7+4+1+(-2)+(-5)+(-8)+(-11)}}}