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Question Number 7163 by Tawakalitu. last updated on 14/Aug/16

Commented by sou1618 last updated on 14/Aug/16

Σ_(a=1) ^4 (Σ_(b=0) ^4 a^b )=Σ_(a=1) ^4 (a^0 +a^1 +a^2 +a^3 +a^4 ) Σ_(a=1) ^4 (1+a+a^2 +a^3 +a^4 ) Σk=(1/2)n(n+1) Σk^2 =(1/6)n(n+1)(2n+1) Σk^3 =(1/4){n(n+1)}^2 Σk^4 =(1/(30))n(n+1)(2n+1)(3n^2 +3n−1) =4+10+(1/6)4(4+1)(2×4+1)+(1/4){4(4+1)}^2 +(1/(30))4(4+1)(8+1)(3×16+3×4−1) =14+30+100+6×59 =498

$$\underset{{a}=\mathrm{1}} {\overset{\mathrm{4}} {\sum}}\left(\underset{{b}=\mathrm{0}} {\overset{\mathrm{4}} {\sum}}{a}^{{b}} \right)=\underset{{a}=\mathrm{1}} {\overset{\mathrm{4}} {\sum}}\left({a}^{\mathrm{0}} +{a}^{\mathrm{1}} +{a}^{\mathrm{2}} +{a}^{\mathrm{3}} +{a}^{\mathrm{4}} \right) \\ $$$$\underset{{a}=\mathrm{1}} {\overset{\mathrm{4}} {\sum}}\left(\mathrm{1}+{a}+{a}^{\mathrm{2}} +{a}^{\mathrm{3}} +{a}^{\mathrm{4}} \right) \\ $$$$\Sigma{k}=\frac{\mathrm{1}}{\mathrm{2}}{n}\left({n}+\mathrm{1}\right) \\ $$$$\Sigma{k}^{\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{6}}{n}\left({n}+\mathrm{1}\right)\left(\mathrm{2}{n}+\mathrm{1}\right) \\ $$$$\Sigma{k}^{\mathrm{3}} =\frac{\mathrm{1}}{\mathrm{4}}\left\{{n}\left({n}+\mathrm{1}\right)\right\}^{\mathrm{2}} \\ $$$$\Sigma{k}^{\mathrm{4}} =\frac{\mathrm{1}}{\mathrm{30}}{n}\left({n}+\mathrm{1}\right)\left(\mathrm{2}{n}+\mathrm{1}\right)\left(\mathrm{3}{n}^{\mathrm{2}} +\mathrm{3}{n}−\mathrm{1}\right) \\ $$$$ \\ $$$$=\mathrm{4}+\mathrm{10}+\frac{\mathrm{1}}{\mathrm{6}}\mathrm{4}\left(\mathrm{4}+\mathrm{1}\right)\left(\mathrm{2}×\mathrm{4}+\mathrm{1}\right)+\frac{\mathrm{1}}{\mathrm{4}}\left\{\mathrm{4}\left(\mathrm{4}+\mathrm{1}\right)\right\}^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{30}}\mathrm{4}\left(\mathrm{4}+\mathrm{1}\right)\left(\mathrm{8}+\mathrm{1}\right)\left(\mathrm{3}×\mathrm{16}+\mathrm{3}×\mathrm{4}−\mathrm{1}\right) \\ $$$$=\mathrm{14}+\mathrm{30}+\mathrm{100}+\mathrm{6}×\mathrm{59} \\ $$$$=\mathrm{498} \\ $$

Commented by Tawakalitu. last updated on 14/Aug/16

Wow, I really appreciate your effort.Thanks so much.

$${Wow},\:{I}\:{really}\:{appreciate}\:{your}\:{effort}.{Thanks}\:{so}\:{much}. \\ $$

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