kindly do the 10th problem

Question

​​​kindly
do the 10th problem

Tejaswi · Accepted Answer

Dear Student,

(a) Let f(x)= 1x+1+2x(x+1)(x+2)+ 3x2(x+1)(x+2)(x+3)+4x3(x+1)(x+2)(x+3)(x+4)
Now,  2x(x+1)(x+2) =x x0x+1- x
x1(x+1)(x+2)= x1x+1- x2(x+1)(x+2)   3x2(x+1)(x+2)(x+3)= x
x1(x+1)(x+2)-x
x2(x+1)(x+2)(x+3)=x2(x+1)(x+2)-x3(x+1)(x+2)(x+3)     
           
       nxn-1(x+1)(x+2)(x+3)
 (x+n)=xn-1(x+1)(x+2)  (x+n-1)-xn(x+1)(x+2)(x+3)
 (x+n)Adding all these terms, we getSo, f(x)=1x+1+x1x+1-xn(x+1)(x+2)(x+3)
 (x+n)=1-xn(x+1)(x+2)(x+3)
 (x+n)

(b)S=  a1(1+a1)+ a2(1+a1)(1+a2) + a3(1+a1)(1+a2)(1+a3)
⇒S= 1-1(1+a1)+1(1+a1)-1(1+a1)(1+a2)+1(1+a1)(1+a2)-1(1+a1)(1+a2)(1+a3)
 1(1+a1)(1+a2)(1+a3) (1+an-1)-1(1+a1)(1+a2)(1+a3)
(1+an)⇒S= 1-1(1+a1)(1+a2)(1+a3) (1+an)

Regards,

​​​kindly do the 10th problem