an question on how to write a summation

Question

So I was doing a problem and reach this step: $\sum\limits_{k=0}^n C_k(a+b\sqrt2)^{k} $. I am not quite sure what is the double summation after perform a polynomial expansion on $(a+b\sqrt2)^{k}$; can someone teach me how to do it? I have a tentative one :$\sum\limits_{k=0}^n \sum\limits_{i=0}^k C_k {n \choose i}a^{k-i}(b \sqrt2)^i$.

Answer 1

You are very close. Your $n \choose i$ comes from the binomial coefficient that is part of the expansion of $(a+b\sqrt 2)^k$ and should therefore be $k \choose i$. You are choosing which of the $k$ terms are $b\sqrt 2$. The formula should be $$\sum\limits_{k=0}^n \sum\limits_{i=0}^k C_k {k \choose i}a^{k-i}(b \sqrt2)^i$$