Binomial problem: Prove sum[k=0,n] (n-choose-k) {sum[i=0,k] (-3)^i (k-choose-i)} = (-1)^n

[elioruzan]

Hi everyone.
I stumbled upon this question and couldn't know how to start.
I need to prove this identity using summation and binomial properties without induction.
Any help would be appreciated!

 

[Dr.Peterson]

Hi everyone.
I stumbled upon this question and couldn't know how to start.
I need to prove this identity using summation and binomial properties without induction.
Any help would be appreciated!View attachment 35031

What does the binomial theorem tell you about the inner sum?

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[elioruzan]

What does the binomial theorem tell you about the inner sum?

View attachment 35035​

Thank you for the quick reply.
Maybe that i can write it down as (1-3)^k?

 

[Dr.Peterson]

Thank you for the quick reply.
Maybe that i can write it down as (1-3)^k?

Yes. Keep going.

 

[elioruzan]

Yes. Keep going.

Still couldn't solve it ?
I know that the sum of even binomial coefficients equals that of the odd but it wasn't really a breakthrough..
Please pinpoint my blind spot ?

 

[Dr.Peterson]

Still couldn't solve it ?
I know that the sum of even binomial coefficients equals that of the odd but it wasn't really a breakthrough..
Please pinpoint my blind spot ?

Please show us what you see! Without knowing what you tried, we can't tell what you missed.

In particular, what does the summation look like when you replace the inner sum with its value? Do the same thing there that you did to find the inner sum.

 

[elioruzan]

Thank you very much. i worked this out.