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Question Number 147988 by puissant last updated on 24/Jul/21

$$\mathrm{La}\mathrm{somme}\mathrm{des}\mathrm{n}\mathrm{premiers}\mathrm{termes}{\mathrm{d}}^{\prime }\mathrm{une}$$$$\mathrm{s}\acute{\mathrm{e}rie}\mathrm{est}\mathrm{donn}\acute{\mathrm{e}}\mathrm{par}{\mathrm{S}}_{\mathrm{n}}={5\mathrm{n}}^2+2\mathrm{n}\mathrm{le}$$$$\mathrm{n}-\mathrm{ieme}\mathrm{terme}\mathrm{de}\mathrm{cette}\mathrm{serie}\mathrm{est}:$$

Answered by Olaf_Thorendsen last updated on 24/Jul/21

$$\mathrm{Soit}{u}_1\mathrm{le}1\mathrm{er}\mathrm{terme}\mathrm{de}\mathrm{la}\mathrm{serie}.$$$$\mathrm{On}\mathrm{a}{u}_1={\mathrm{S}}_1=5{\left(1\right)}^2+2\left(1\right)=7$$$$$$$$\mathrm{Soit}{u}_k\mathrm{les}\mathrm{termes}\mathrm{de}\mathrm{la}\mathrm{serie}.$$$${\mathrm{S}}_n=\underset{k=1}{\sum ^n}{u}_k=5n^2+2n$$$$\mathrm{Et}\mathrm{donc},\mathrm{pour}n⩾2,\mathrm{on}\mathrm{a}:$$$${\mathrm{S}}_{n-1}=\underset{k=1}{\sum ^{n-1}}{u}_k=5{(n-1)}^2+2(n-1)$$$$\mathrm{Le}{n}^{ieme}\mathrm{terme}\mathrm{vaut}:$$$$u_n={\mathrm{S}}_n-S_{n-1}=5n^2+2n-5{(n-1)}^2-2(n-1)$$$$u_n=10n-3$$$$$$$$\mathrm{A}\mathrm{partir}\mathrm{de}{u}_n,\mathrm{on}\mathrm{peut}\mathrm{alors}\mathrm{retrouver}$$$$\mathrm{reciproquement}{\mathrm{l}}^{\prime }\mathrm{expression}\mathrm{de}{\mathrm{S}}_n:$$$${\mathrm{S}}_n=\underset{k=1}{\sum ^n}{u}_k=\underset{k=1}{\sum ^n}(10k-3)=10\underset{k=1}{\sum ^n}k-\underset{k=1}{\sum ^n}3$$$${\mathrm{S}}_n=10\frac{n(n+1)}{2}-3n=5n^2+2n$$

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