a.
\(u_n=u_1+\left(n-1\right)d\Rightarrow4+\left(n-1\right).\left(-3\right)=-41\)
\(\Rightarrow n=16\)
b.
\(u_n=5n-7\Rightarrow u_1=5.1-7=-2\)
\(S_n=\dfrac{n\left(u_1+u_n\right)}{2}=817\Rightarrow\dfrac{n\left(-2+5n-7\right)}{2}=817\)
\(\Rightarrow5n^2-9n-1634=0\Rightarrow\left[{}\begin{matrix}n=19\\n=-\dfrac{86}{5}\left(loại\right)\end{matrix}\right.\)
c.
\(u_4=u_1+3d\) ; \(u_{14}=u_1+13d\)
\(\Rightarrow u_{14}-u_4=13d+u_1-\left(u_1+3d\right)=10d\)
\(\Rightarrow18-\left(-12\right)=10d\Rightarrow d=3\)
d.
\(\left\{{}\begin{matrix}u_2+u_3=20\\u_5+u_7=-29\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}\left(u_1+d\right)+\left(u_1+2d\right)=20\\\left(u_1+4d\right)+\left(u_1+6d\right)=-29\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}2u_1+3d=20\\2u_1+10d=-29\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}u_1=\dfrac{41}{2}\\d=-7\end{matrix}\right.\)
\(\Rightarrow u_{12}=u_1+11d=-\dfrac{113}{2}\)
