Answer
24 terms
Work Step by Step
We are given the arithmetic sequence:
$a_1=11$
$d=3$
$S_n=1092$
We determine $n$ solving the equation:
$S_n=\dfrac{n(2a_1+(n-1)d)}{2}$
$1092=\dfrac{n(2(11)+(n-1)(3)}{2}$
$2(1092)=n(22+3n-3)$
$2184=n(3n+19)$
$3n^2+19n-2184=0$
$n=\dfrac{-19\pm\sqrt{19^2-4(3)(-2184)}}{2(3)}=\dfrac{-19\pm 163}{6}$
$n_1=\dfrac{-19-163}{6}=-\dfrac{182}{6}=-\dfrac{91}{3}$
$n_2=\dfrac{-19+163}{6}=\dfrac{144}{6}=24$
As $n$ must be a positive integer, the only solution is:
$n=24$
