If... $color(white)x^(x + 1)C_3 - ^(x - 1)C_3 = 16x = ?$

Question

1875 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-11-22T08:58:44

$x=5$

$C_3^(x+1)=((x+1)x(x-1))/(1.2.3)$

and $C_3^(x-1)=((x-1)(x-2)(x-3))/(1.2.3)$

Therefore $C_3^(x+1)-C_3^(x-1)$

= $((x+1)x(x-1))/(1.2.3)-((x-1)(x-2)(x-3))/(1.2.3)$

= $(x-1)/6{x(x+1)-(x-2)(x-3)}$

= $(x-1)/6{x^2+x-(x^2-5x+6)}$

= $(x-1)/6(6x-6)$

= $(x-1)^2$

Hence $C_3^(x+1)-C_3^(x-1)=16x$ is equivalent to

$(x-1)^2=16$

or $x-1=+-4$

i.e. $x=5$ or $-3$

and if we consider only as a positive integer $x=5$

If... color(white)x^(x + 1)C_3 - ^(x - 1)C_3 = 16x = ?color(white)x^(x + 1)C_3 - ^(x - 1)C_3 = 16x = ?