Question

Find the sum $3{{}^{n}C}_0-8{{}^{n}C}_1+13{{}^{n}C}_2-18\times {{}^{n}C}_3+....$

Answer 1

$sum={\Sigma }_{r=0}^n\left(5r\right)(-1{)}^r{C}_r+3{\Sigma }_{r=0}^n(-1{)}^r{C}_r$

$=5(-n)(1-1{)}^{n-1}+3(1-1{)}^n=0+0=0$