Question

Find the 12th term of the arithmetic sequence -2x-8, 2x-3, 6x+2

Answer 1

to get the common difference on any arithmetic sequence, we only need to get the difference from a following term and its previous term, so in this case let use the 1st and 2nd, and do (2nd) - (1st) to get it.

`\stackrel{2nd}{(2x-3)}~~ - ~~\stackrel{1st}{(-2x-8)}\implies 2x-3+2x+8\implies \stackrel{ \textit{common difference} }{4x+5} \\\\[-0.35em] ~\dotfill\\\\ n^(th)\textit{ term of an arithmetic sequence} \\\\ a_n=a_1+(n-1)d\qquad \begin{cases} a_n=n^(th)\ term\\ n=\textit{term position}\\ a_1=\textit{first term}\\ d=\textit{common difference}\\[-0.5em] \hrulefill\\ d=4x+5\\ a_1=-2x-8\\ n=12 \end{cases}`

`a_(12)=\stackrel{ a_1 }{-2x-8}+(12-1)\stackrel{ d }{(4x+5)}\implies a_(12)=-2x-8+(11)(4x+5) \\\\\\ a_(12)=-2x-8+44x+55\implies \boxed{a_(12)=42x+47}`