Question Number 50362 by prof Abdo imad last updated on 16/Dec/18
![calculate S_1 =Σ_(k=0) ^n C_n ^k S_2 =Σ_(k=0) ^([(n/2)]) C_n ^(2k) S_3 = Σ_(k=0) ^([(n/3)]) C_n ^(3k)](Q50362.png)
$${calculate}\:{S}_{\mathrm{1}} =\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{k}} \\ $$$${S}_{\mathrm{2}} =\sum_{{k}=\mathrm{0}} ^{\left[\frac{{n}}{\mathrm{2}}\right]} \:\:{C}_{{n}} ^{\mathrm{2}{k}} \\ $$$${S}_{\mathrm{3}} =\:\sum_{{k}=\mathrm{0}} ^{\left[\frac{{n}}{\mathrm{3}}\right]} \:{C}_{{n}} ^{\mathrm{3}{k}} \\ $$