This is an elementary question asked by the beginner at the frontdoor of higher mathematics.
In Burton's Elementary Number Theory ( seventh edition, page 9) I find the following equality written under the (induction) hypothesis that the binomial formula works for $( a+b)^m$:
$$\Large b \space (a + b)^m = b \sum_{j=0}^{m} \binom{m}{j} a^{m-j} b^{j} = \sum_{j=0}^{m} \binom{m}{j} a^{m-j} b^{j+1}$$
This is fine to me. After that, the author operates an index change : $ j= k-1 $ or , equivalently $ k = j+1$, which yields :
$$\Large \sum_{k=1}^{m} \binom{m}{k-1} a^{m+1-k} b^k + b^{m+1}$$.
I understand well why , in the binomial coefficient $ j$ becomes $ k-1$, why the power of $a$ becomes $ a-(k-1) = a+1-k$ , why the power of the first occurrence of $b$ becomes $ k $ ( since $j+1 = k$).
What I do not understand is the appearing of a second term in the scope of the sum sign , namely the term $ b ^ {m+1}$.
What do I miss?
Thaks for any hint.