Question Number 64871 by Rio Michael last updated on 22/Jul/19 | ||
$${any}\:{hint}\:{about}\:{how}\:{to}\:{prove}\:{by}\:{induction}\:{in}\:{the}\:{Sigma}\:{notion}\:{topic}? \\ $$$${like}\:{in}\:\Sigma \\ $$ | ||
Commented by mathmax by abdo last updated on 22/Jul/19 | ||
$${you}\:{question}\:{is}\:{not}\:{clear}.... \\ $$ | ||
Commented by Rio Michael last updated on 22/Jul/19 | ||
$${help}\:{me}\:{know}\:{how}\:{to}\:{prove}\:{by}\:{mathematical}\:{induction}\:{when}\:{dealing}\:{with}\:{summation} \\ $$$$\Sigma \\ $$ | ||
Commented by mathmax by abdo last updated on 22/Jul/19 | ||
$${i}\:{dont}\:{understand}\:{what}\:{you}\:{want}\:{but}\:{i}\:{give}\:{some}\:{properties}\:{of} \\ $$$${of}\:\:\Sigma\:\:\:\:{if}\:{we}\:{have}\:\:\left({x}_{{i}} \right)\:_{\mathrm{1}\leqslant{i}\leqslant{n}} \:\:{and}\:\left({y}_{{i}} \right)_{{i}\in\left[\mathrm{0},{n}\right]} \:\mathrm{2}\:{sequences}\:{of}\:{reals} \\ $$$${numbers}\:\left({or}\:{complex}\right)\:\:{the}\:{sum}\:{x}_{\mathrm{1}} \:+{x}_{\mathrm{2}} +....+{x}_{{k}} \:{is}\:{writen} \\ $$$$\sum_{{i}=\mathrm{1}} ^{{k}} \:{x}_{{i}} \:\:\:\:{also}\:{we}\:{have}\:\sum_{{i}=\mathrm{1}} ^{{n}} \:{x}_{{i}} \:\overset{−} {+}\sum_{{i}=\mathrm{1}} ^{{n}} \:{y}_{{i}} =\sum_{{i}=\mathrm{1}} ^{{n}} \left({x}_{{i}} \:\overset{−} {+}{y}_{{i}} \right) \\ $$$$\sum_{{i}=\mathrm{1}} ^{{n}} \:\lambda\:{x}_{{i}} =\lambda\:\sum_{{i}=\mathrm{1}} ^{{n}} \:{x}_{{i}} \:\:\left(\lambda\:\in{R}\:{or}\:{C}\right) \\ $$$$\sum_{{i}=\mathrm{1}} ^{{n}} \left(\mathrm{1}\right)\:={n}\:\:\:,\sum_{{i}=\mathrm{0}} ^{{n}} \left(\mathrm{1}\right)\:={n}+\mathrm{1}\:,\sum_{{i}=\mathrm{1}} ^{{n}} \lambda\:={n}\lambda \\ $$$$\sum_{{i}=\mathrm{0}} ^{{n}} \:\lambda\:=\left({n}+\mathrm{1}\right)\lambda\:\:\:\:,\:\sum_{{i}} \:\sum_{{j}} \:{x}_{{ij}} =\sum_{{j}} \sum_{{i}} {x}_{{ij}} \:\:.... \\ $$$$ \\ $$ | ||
Commented by Rio Michael last updated on 22/Jul/19 | ||
$${well}\:{thanks}\:{for}\:{that} \\ $$ | ||
Commented by mathmax by abdo last updated on 23/Jul/19 | ||
$${you}\:{are}\:{welcome}. \\ $$ | ||