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Find-the-sum-of-the-series-2-5-8-12-n-




Question Number 10369 by Tawakalitu ayo mi last updated on 05/Feb/17
Find the sum of the series  2 + 5 + 8 + 12 ... n
$$\mathrm{Find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{series} \\ $$$$\mathrm{2}\:+\:\mathrm{5}\:+\:\mathrm{8}\:+\:\mathrm{12}\:…\:\mathrm{n}\: \\ $$
Commented by mrW1 last updated on 06/Feb/17
the n−th term of the series is not  clear. please check.
$${the}\:{n}−{th}\:{term}\:{of}\:{the}\:{series}\:{is}\:{not} \\ $$$${clear}.\:{please}\:{check}. \\ $$
Answered by arge last updated on 06/Feb/17
d=3  a_1 =2  a_n =?    a_n =a_1 +(n−1)d    a_n =2+3n−3    a_n =3n−1    a_n =((2S_n )/n) −a_1       3n^2 −n=2S_n −a_1 n    3n^2 +n=2S_n     S_n =((n(3n+1))/2)
$${d}=\mathrm{3} \\ $$$${a}_{\mathrm{1}} =\mathrm{2} \\ $$$${a}_{{n}} =? \\ $$$$ \\ $$$${a}_{{n}} ={a}_{\mathrm{1}} +\left({n}−\mathrm{1}\right){d} \\ $$$$ \\ $$$${a}_{{n}} =\mathrm{2}+\mathrm{3}{n}−\mathrm{3} \\ $$$$ \\ $$$${a}_{{n}} =\mathrm{3}{n}−\mathrm{1} \\ $$$$ \\ $$$${a}_{{n}} =\frac{\mathrm{2}{S}_{{n}} }{{n}}\:−{a}_{\mathrm{1}} \\ $$$$ \\ $$$$ \\ $$$$\mathrm{3}{n}^{\mathrm{2}} −{n}=\mathrm{2}{S}_{{n}} −{a}_{\mathrm{1}} {n} \\ $$$$ \\ $$$$\mathrm{3}{n}^{\mathrm{2}} +{n}=\mathrm{2}{S}_{{n}} \\ $$$$ \\ $$$${S}_{{n}} =\frac{{n}\left(\mathrm{3}{n}+\mathrm{1}\right)}{\mathrm{2}} \\ $$$$ \\ $$

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