Let , ( ) be the values of m , for which the equations x+y+z=1 ; x+2 y+4 z= m and x+4 y+10 z=m^2 have infinitely many solutions. Then the value of _ n=1 ¹⁰ (n^ +n^ ) is equal to :

Step-by-step solution

aligned & = | array llc 1 & 1 & 1 1 & 2 & 4 1 & 4 & 10 array |=1(20-16)-1(10-4)+1(4-2) & =4-6+2=0 aligned For infinite solutions aligned & _x= _y= _z=0 & m^2-3 x+2=0 & m=1,2 & =1, =2 & _ n=1 ¹⁰ (n^ +n^ )= _ n=1 ¹⁰ n^1+ _ n=1 ¹⁰ n^2 &= 10(11) 2 + 10(11)(21) 6 &= 55+385 &= 440 aligned