Question

Find the first four terms of:

a) An arithmetic series with first term of 3 and sum to five terms is 75

Answer 1

`\textit{sum of a finite arithmetic sequence} \\\\ S_n=\cfrac{n(a_1+a_n)}{2}\qquad \begin{cases} a_n=n^(th)\ term\\ n=\textit{last term's}\\ \qquad position\\ a_1=\textit{first term}\\[-0.5em] \hrulefill\\ a_1=3\\ n=5\\ S_5=75 \end{cases} \implies \begin{array}{llll} S_5=\cfrac{5(a_1+a_5)}{2} \\\\\\ 75=\cfrac{5(3+a_5)}{2} \end{array} \\\\\\ 150=5(3+a_5)\implies \cfrac{150}{5}=3+a_5\implies 30=3+a_5\implies \boxed{27=a_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\\ n=5\\ a_5=27\\ a_1=3 \end{cases}\implies 27=3+(5-1)d \\\\\\ 24=(5-1)d\implies 24=4d\implies \cfrac{24}{4}=d\implies \boxed{6=d} \\\\[-0.35em] ~\dotfill\\\\ ~\hfill \stackrel{\textit{\LARGE Terms}}{3~~,~~\stackrel{3+6}{9}~~,~~\stackrel{9+6}{15}~~,~~\stackrel{15+6}{21}}~\hfill`