If the roots of x^5-a x^4+b x^3-c x^2+d x-1=0 are all positive such that their arithmetic mean and geometric mean are equal, then a+b+c+d=

Step-by-step solution

Given equation is x^5-a x^4+b x-c x+d x-1=0 . Here, geometrical mean and arithmetic mean are equal then all the roots are equal. Let is the root of the equation. + + + + 5 = ( ^5 )^ 1 / 5 Here, sum of all roots is a . gathered a 5 = a=5 gathered ( )^ 1 / 5 = -e a =-(-1)=1 aligned & ^ 5 1 5 =1 & =1 aligned From (i), a=5 . Now, aligned & + + + . .+ (10 times )=b & 10 ^2=b & b=10 & + + . .+10 times =c & 10 ^3=c & c=10 & + + +5 times =d & 5 ^4=d & d=5 aligned Therefore, a+d+c+b=5+5+10+10=30 .