TRY THIS

Show that
$C 0 - C 1 ( 1 + n x 1 + x ) + C 2 ( (1 + n x) 2 1 + 2 x ) - C 3 ( (1 + n x) 3 1 + 3 x ) + . . . . . . . . . = 0$

1 Answers

Check if i'm wrong anywhere..

$\sum r = 0 n (- 1) r C r n (1 + n x) r 1 + r x $

= $\sum r = 0 n (- 1) r C r n (1 + n x) r 1 $ + $\sum r = 0 n (- 1) r r n C r - 1 n - 1 (1 + n x) r r x $

Solving further...we get,

$(1 - 1 + n x 1 ) n - 1 + n x n x (1 - 1 + n x 1 ) n - 1 $

Which gives,

$( 1 + n x n x ) n - ( 1 + n x n x ) n = 0$

I skipped a step

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