Question

Find the sum of
$3{\cdot }^n{C}_0-8\cdot {}^n{C}_1+13\cdot {}^n{C}_2-18\cdot {}^n{C}_3+....+$ upto$(n+1)$terms.

Answer 1

This is Q.7 (b) .Let $ denote the sum of the series .
General term of the series is given by
$T_r=(-1{)}^r(3+5r){}^n{C}_r,(nthtermofA.P.)$
Where r = 0,1,2,......n
$\therefore S={\sum }_{r=0}^n(-1{)}^r(3+5r){}^n{C}_r$
$=3{\sum }_{r=0}^n(-1{)}^r{}^n{C}_r+5{\sum }_{r=0}^n(-1{)}^{r}r{}^n{C}_r$
$=3(C_0-C_1+C_2-C_3+C_4.....)+5(-C_1+2C_2-3C_3+4C_4)$
$=0+0$