Question Number 10369 by Tawakalitu ayo mi last updated on 05/Feb/17
$$\mathrm{Find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{series} \\ $$$$\mathrm{2}\:+\:\mathrm{5}\:+\:\mathrm{8}\:+\:\mathrm{12}\:...\:\mathrm{n}\: \\ $$
Commented by mrW1 last updated on 06/Feb/17
$${the}\:{n}−{th}\:{term}\:{of}\:{the}\:{series}\:{is}\:{not} \\ $$$${clear}.\:{please}\:{check}. \\ $$
Answered by arge last updated on 06/Feb/17
$${d}=\mathrm{3} \\ $$$${a}_{\mathrm{1}} =\mathrm{2} \\ $$$${a}_{{n}} =? \\ $$$$ \\ $$$${a}_{{n}} ={a}_{\mathrm{1}} +\left({n}−\mathrm{1}\right){d} \\ $$$$ \\ $$$${a}_{{n}} =\mathrm{2}+\mathrm{3}{n}−\mathrm{3} \\ $$$$ \\ $$$${a}_{{n}} =\mathrm{3}{n}−\mathrm{1} \\ $$$$ \\ $$$${a}_{{n}} =\frac{\mathrm{2}{S}_{{n}} }{{n}}\:−{a}_{\mathrm{1}} \\ $$$$ \\ $$$$ \\ $$$$\mathrm{3}{n}^{\mathrm{2}} −{n}=\mathrm{2}{S}_{{n}} −{a}_{\mathrm{1}} {n} \\ $$$$ \\ $$$$\mathrm{3}{n}^{\mathrm{2}} +{n}=\mathrm{2}{S}_{{n}} \\ $$$$ \\ $$$${S}_{{n}} =\frac{{n}\left(\mathrm{3}{n}+\mathrm{1}\right)}{\mathrm{2}} \\ $$$$ \\ $$