Solving the alternating sum of binomial coefficients using telescoping series

[a8281333]

I havn't done this in a long time! And apparently I should know this easy, it sort of looks like a proof by induction, which I havn't done before and I am frantically trying to learn!

Show that for each integer n the alternating sum of binomial coefficients:

1 - (n) + ... + (-1)^k(n) + ... + (-1)^(n-1)( n ) + (-1)^n
...(1)......(k)...... ...(n-1)
is zero. What is the value of the sum

(so what I've done here is started with an "inductive basis of n=1 which kind of suggests it goes to zero but without the appropriate conciseness)

1 + (n) + ... + (n) + ... + ( n ) +1
...(1)...(k)...(n-1)

I understand the layout is a bit rubbish but I hope you can fathom it!
Any help would be greatly appreciated!

UPDATE! After a bit of research, am I correct in assuming this is a telescoping series?

 

[ystael]

"Binomial" is the key word here. Try to find a clever way to manufacture what you see in front of you by expanding some expression.