Question

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

• A. zero
• B. $n+1$
• C. $n-1$
• D. $(n+1{)}^2$

Answer 1

The correct option is A zero

${(1-x)}^n=C_0-C_{1}x+C_2{x}^2+....+{(-1)}^n{C}_n{x}^n$ ....(1)
${(1-x^5)}^n=C_0-C_1{x}^5+C_2{x}^{10}+....+{(-1)}^n{C}_n{x}^{5n}$
Differentiating wrt x
$-5n{x}^4{(1-x^5)}^{n-1}={-5C}_1{x}^4+10C_2{x}^9+15C_2{x}^{14}+....+{(-1)}^n{C}_n{x}^{5n}$
Multiplying (1) by 3 and then adding with (2):
$3{(1-x)}^n-5n{x}^4{(1-x^5)}^{n-1}=3.{{}^{n}C}_0-8{{}^{n}C}_1{x}^4+13{{}^{n}C}_2{x}^9-18{{}^{n}C}_3{x}^{14}....$
now, put $x=0$
$0=3{{}^{n}C}_0-8{{}^{n}C}_1+13{{}^{n}C}_2-18{{}^{n}C}_3....$