Question Number 69572 by Ajao yinka last updated on 25/Sep/19
Answered by mind is power last updated on 25/Sep/19
$${Let}\:{S}_{\mathrm{0}} =\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\left(_{\mathrm{5}{k}} ^{\mathrm{5}{n}} \right)\:\:\:,{S}_{\mathrm{1}} =\underset{{k}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\sum}}\left(_{\mathrm{5}{k}+\mathrm{1}} ^{\mathrm{5}{n}} \right),{S}_{\mathrm{2}} =\underset{{k}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\sum}}\left(_{\mathrm{5}{k}+\mathrm{2}} ^{\mathrm{5}{n}} \right),{S}_{\mathrm{3}} =\underset{{k}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\sum}}\left(_{\mathrm{5}{k}+\mathrm{3}} ^{\mathrm{5}{n}} \right),{S}_{\mathrm{4}} =\underset{{k}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\sum}}\left(_{\mathrm{5}{k}+\mathrm{4}} ^{\mathrm{5}{n}} \right) \\ $$$$\Rightarrow{S}_{\mathrm{0}} +{S}_{\mathrm{1}} +{S}_{\mathrm{2}} +{S}_{\mathrm{3}} +{S}_{\mathrm{4}} =\underset{{k}=\mathrm{0}} {\overset{\mathrm{5}{n}} {\sum}}\left(_{{k}} ^{\mathrm{5}{n}} \right)=\mathrm{2}^{\mathrm{5}{n}} ……\mathrm{1} \\ $$$${Let}\:\:{f}=\left(\mathrm{1}+{e}^{\frac{\mathrm{2}{i}\pi}{\mathrm{5}}} \right)^{\mathrm{5}{n}} ={e}^{{in}\pi} \left({e}^{\frac{{i}\pi}{\mathrm{5}}} +{e}^{\frac{−{i}\pi}{\mathrm{5}}} \right)^{\mathrm{5}{n}} =\left(−\mathrm{1}\right)^{{n}} \ast\mathrm{2}^{\mathrm{5}{n}} {cos}^{\mathrm{5}{n}} \left(\frac{\pi}{\mathrm{5}}\right) \\ $$$${f}=\left(\mathrm{1}+{e}^{\frac{\mathrm{2}{i}\pi}{\mathrm{5}}} \right)^{\mathrm{5}{n}} =\underset{{k}=\mathrm{0}} {\overset{\mathrm{5}{n}} {\sum}}\left(_{{k}} ^{\mathrm{5}{n}} \right){e}^{\frac{\mathrm{2}{ik}\pi}{\mathrm{5}}} =\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\left(_{\mathrm{5}{k}} ^{\mathrm{5}{n}} \right)+\underset{{k}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\sum}}\left(_{\mathrm{5}{k}+\mathrm{1}} ^{\mathrm{5}{n}} \right){e}^{\frac{\mathrm{2}{i}\pi}{\mathrm{5}}} +\underset{{k}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\sum}}\left(_{\mathrm{5}{k}+\mathrm{2}} ^{\mathrm{5}{n}} \right){e}^{\frac{\mathrm{4}{i}\pi}{\mathrm{5}}} +\underset{{k}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\sum}}\left(_{\mathrm{5}{k}+\mathrm{3}} ^{\mathrm{5}{n}} \right){e}^{\frac{\mathrm{6}{i}\pi}{\mathrm{5}}} +\underset{{k}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\sum}}\left(_{\mathrm{5}{k}+\mathrm{4}} ^{\mathrm{5}{n}} \right){e}^{\frac{\mathrm{8}{i}\pi}{\mathrm{5}}} \\ $$$$={S}_{\mathrm{0}} +{e}^{\frac{\mathrm{2}{i}\pi}{\mathrm{5}}} {S}_{\mathrm{1}} +{e}^{\frac{\mathrm{4}{i}\pi}{\mathrm{5}}} {S}_{\mathrm{2}} +{e}^{\frac{−\mathrm{4}{i}\pi}{\mathrm{5}}} {S}_{\mathrm{3}} +{e}^{\frac{−\mathrm{2}{i}\pi}{\mathrm{5}}} {S}_{\mathrm{4}} =\left(−\mathrm{1}\right)^{{n}} \mathrm{2}^{\mathrm{5}{n}} {cos}^{\mathrm{5}{n}} \left(\frac{\pi}{\mathrm{5}}\right) \\ $$$${We}\:{Tack}\:{real}\:{and}\:{Im}\:{part}\Rightarrow \\ $$$$\:\:{S}_{\mathrm{0}} +{cos}\left(\frac{\mathrm{2}\pi}{\mathrm{5}}\right)\left({S}_{\mathrm{1}} +{S}_{\mathrm{4}} \right)+{cos}\left(\frac{\mathrm{4}\pi}{\mathrm{5}}\right)\left({S}_{\mathrm{2}} +{S}_{\mathrm{3}} \right)=\left(−\mathrm{1}\right)^{{n}} \mathrm{2}^{\mathrm{5}{n}} ×{cos}^{\mathrm{5}{n}} \left(\frac{\pi}{\mathrm{5}}\right)…\mathrm{2} \\ $$$${sin}\left(\frac{\mathrm{2}\pi}{\mathrm{5}}\right)\left({S}_{\mathrm{1}} −{S}_{\mathrm{4}} \right)+{sin}\left(\frac{\mathrm{4}\pi}{\mathrm{5}}\right)\left({S}_{\mathrm{2}} −{S}_{\mathrm{3}} \right)=\mathrm{0}…\mathrm{3} \\ $$$${Now}\:{let}\:{Evaluat}\:\left(\mathrm{1}+{e}^{\frac{\mathrm{4}{i}\pi}{\mathrm{5}}} \right)^{\mathrm{5}{n}} =\left({e}^{\frac{\mathrm{2}{i}\pi}{\mathrm{5}}} \left({e}^{\frac{\mathrm{2}{i}\pi}{\mathrm{5}}} +{e}^{\frac{−\mathrm{2}{i}\pi}{\mathrm{5}}} \right)\right)^{\mathrm{5}{n}} ={e}^{\mathrm{2}{in}\pi} \left(\mathrm{2}{cos}\left(\frac{\mathrm{2}\pi}{\mathrm{5}}\right)\right)^{\mathrm{5}{n}} =\mathrm{2}^{\mathrm{5}{n}} {cos}^{\mathrm{5}{n}} \left(\frac{\mathrm{2}\pi}{\mathrm{5}}\right) \\ $$$$\left(\mathrm{1}+{e}^{\frac{\mathrm{4}{i}\pi}{\mathrm{5}}} \right)^{\mathrm{5}{n}} ={S}_{\mathrm{0}} +{e}^{\frac{\mathrm{4}{i}\pi}{\mathrm{5}}} {S}_{\mathrm{1}} +{e}^{\frac{−\mathrm{2}{i}\pi}{\mathrm{5}}} {S}_{\mathrm{2}} +{e}^{\frac{\mathrm{2}{i}\pi}{\mathrm{5}}} {S}_{\mathrm{3}} +{e}^{−\frac{\mathrm{4}{i}\pi}{\mathrm{5}}} {S}_{\mathrm{4}} =\mathrm{2}^{\mathrm{5}{n}} {cos}^{\mathrm{5}{n}} \left(\frac{\mathrm{2}\pi}{\mathrm{5}}\right) \\ $$$$\Rightarrow \\ $$$${S}_{\mathrm{0}} +{cos}\left(\frac{\mathrm{4}\pi}{\mathrm{5}}\right)\left({S}_{\mathrm{1}} +{S}_{\mathrm{4}} \right)+{cos}\left(\frac{\mathrm{2}\pi}{\mathrm{5}}\right)\left({S}_{\mathrm{2}} +{S}_{\mathrm{3}} \right)=\mathrm{2}^{\mathrm{5}{n}} \mathrm{cos}\:^{\mathrm{5}{n}} \left(\frac{\mathrm{2}\pi}{\mathrm{5}}\right)…\mathrm{4} \\ $$$${and}\:\left({S}_{\mathrm{1}} −{S}_{\mathrm{4}} \right){sin}\left(\frac{\mathrm{4}\pi}{\mathrm{5}}\right)+\left({S}_{\mathrm{3}} −{S}_{\mathrm{2}} \right){sin}\left(\frac{\mathrm{2}\pi}{\mathrm{5}}\right)=\mathrm{0}\:\:\:\:……\mathrm{5} \\ $$$$\mathrm{3\&5}\Leftrightarrow \\ $$$$\begin{pmatrix}{{sin}\left(\frac{\mathrm{2}\pi}{\mathrm{5}}\right)\:\:\:\:\:\:\:\:{sin}\left(\frac{\mathrm{4}\pi}{\mathrm{5}}\right)}\\{{sin}\left(\frac{\mathrm{4}\pi}{\mathrm{5}}\right)\:\:\:\:−{sin}\left(\frac{\mathrm{2}\pi}{\mathrm{5}}\right)}\end{pmatrix}\:\:\:\:\begin{pmatrix}{\left({S}_{\mathrm{1}} −{S}_{\mathrm{4}} \right)}\\{\left({S}_{\mathrm{2}} −{S}_{\mathrm{3}} \right)}\end{pmatrix}=\begin{pmatrix}{\mathrm{0}}\\{\mathrm{0}}\end{pmatrix} \\ $$$${Det}\begin{bmatrix}{{sin}\left(\frac{\mathrm{2}\pi}{\mathrm{5}}\right)\:\:\:\:\:\:\:{sin}\left(\frac{\mathrm{4}\pi}{\mathrm{5}}\right)}\\{{sin}\left(\frac{\mathrm{4}\pi}{\mathrm{5}}\right)\:\:−{sin}\left(\frac{\mathrm{2}\pi}{\mathrm{5}}\right)}\end{bmatrix}\neq\mathrm{0}\Rightarrow{S}_{\mathrm{1}} ={S}_{\mathrm{4}} \&{S}_{\mathrm{2}} ={S}_{\mathrm{3}} \\ $$$${Substitution}\:{S}_{\mathrm{4}} {by}\:{S}_{\mathrm{1}} {and}\:{S}_{\mathrm{3}} {byS}_{\mathrm{2}} \:{Our}\:{equation}\:{Becom} \\ $$$$\begin{cases}{{S}_{\mathrm{0}} +\mathrm{2}{S}_{\mathrm{1}} +\mathrm{2}{S}_{\mathrm{2}} =\mathrm{2}^{\mathrm{5}{n}} ….{a}}\\{{S}_{\mathrm{0}} +\mathrm{2}{S}_{\mathrm{1}} {cos}\left(\frac{\mathrm{2}\pi}{\mathrm{5}}\right)+\mathrm{2}{S}_{\mathrm{2}} {cos}\left(\frac{\mathrm{4}\pi}{\mathrm{5}}\right)=\left(−\mathrm{1}\right)^{{n}} \mathrm{2}^{\mathrm{5}{n}} \mathrm{cos}\:^{\mathrm{5}{n}} \left(\frac{\pi}{\mathrm{5}}\right)..{b}}\end{cases} \\ $$$${and}\:{S}_{\mathrm{0}} +\mathrm{2}{S}_{\mathrm{1}} {cos}\left(\frac{\mathrm{4}\pi}{\mathrm{5}}\right)+\mathrm{2}{S}_{\mathrm{2}} \mathrm{cos}\:\left(\frac{\mathrm{2}\pi}{\mathrm{5}}\right)=\mathrm{2}^{\mathrm{5}{n}} {cos}^{\mathrm{5}{n}} \left(\frac{\mathrm{2}\pi}{\mathrm{5}}\right)…{c} \\ $$$${after}\:{See}\:{that}\:\mathrm{4}{cos}^{\mathrm{2}} \left(\frac{\mathrm{2}\pi}{\mathrm{5}}\right)+\mathrm{2}{cos}\left(\frac{\mathrm{2}\pi}{\mathrm{5}}\right)−\mathrm{1}=\mathrm{0}=\mathrm{2}{cos}\left(\frac{\mathrm{4}\pi}{\mathrm{5}}\right)+\mathrm{2}{cos}\left(\frac{\mathrm{2}\pi}{\mathrm{5}}\right)+\mathrm{1} \\ $$$$ \\ $$$$\Rightarrow{a}×\frac{\mathrm{1}}{\mathrm{2}}+{b}+{c}=\frac{\mathrm{5}}{\mathrm{2}}{S}_{\mathrm{0}} +{S}_{\mathrm{1}} \left(\mathrm{1}+\mathrm{2}{cos}\left(\frac{\mathrm{2}\pi}{\mathrm{5}}\right)+\mathrm{2}{cos}\left(\frac{\mathrm{4}\pi}{\mathrm{5}}\right)\right)+{S}_{\mathrm{2}} \left(\mathrm{1}+\mathrm{2}{cos}\left(\frac{\mathrm{2}\pi}{\mathrm{5}}\right)+{cos}\left(\frac{\mathrm{4}\pi}{\mathrm{5}}\right)\right)=\mathrm{2}^{\mathrm{5}{n}} \left(\frac{\mathrm{1}}{\mathrm{2}}+\left(−\mathrm{1}\right)^{{n}} {cos}^{\mathrm{5}{n}} \left(\frac{\pi}{\mathrm{5}}\right)+{cos}^{\mathrm{5}{n}} \left(\frac{\mathrm{2}\pi}{\mathrm{5}}\right)\right) \\ $$$$\Rightarrow\frac{\mathrm{5}}{\mathrm{2}}{S}_{\mathrm{0}} =\mathrm{2}^{\mathrm{5}{n}} \left(\frac{\mathrm{1}}{\mathrm{2}}+\left(−\mathrm{1}\right)^{{n}} {cos}^{\mathrm{5}{n}} \left(\frac{\pi}{\mathrm{5}}\right)+{cos}^{\mathrm{5}{n}} \left(\frac{\mathrm{2}\pi}{\mathrm{5}}\right)\right) \\ $$$$\Rightarrow{S}_{\mathrm{0}} =\frac{\mathrm{2}^{\mathrm{5}{n}} }{\mathrm{5}}\left(\mathrm{1}+\mathrm{2}\left(\left(−\mathrm{1}\right)^{{n}} {cos}^{\mathrm{5}{n}} \left(\frac{\pi}{\mathrm{5}}\right)+{cos}^{\mathrm{5}{n}} \left(\frac{\mathrm{2}\pi}{\mathrm{5}}\right)\right)\right) \\ $$$$\Leftrightarrow\underset{{k}=\mathrm{0}} {\overset{\mathrm{5}{n}} {\sum}}\begin{pmatrix}{\mathrm{5}{n}}\\{\mathrm{5}{k}}\end{pmatrix}=\frac{\mathrm{2}^{\mathrm{5}{n}} }{\mathrm{5}}\left(\mathrm{1}+\mathrm{2}\left(\left(−\mathrm{1}\right)^{{n}} \mathrm{cos}\:^{\mathrm{5}{n}} \left(\frac{\pi}{\mathrm{5}}\right)+\mathrm{cos}\:^{\mathrm{5}{n}} \left(\frac{\mathrm{2}\pi}{\mathrm{5}}\right)\right)\right) \\ $$$$ \\ $$
Commented by Ajao yinka last updated on 26/Sep/19
$${nice}\:{one} \\ $$
Commented by otchereabdullai@gmail.com last updated on 29/Sep/19
$$\mathrm{powerful} \\ $$