(Feller Volume 1, Q.44, P. 173) Let $$B(k; n, p) = \sum_{v=0}^k b(v; n,p)$$ be the probability of at most $k$ successes in $n$ trials. Then $$B(k; n+1, p ) = B(k; n,p) - p b(k; n,p),$$ $$B(k+1 ; n+1, p) = B(k; n,p) + qb(k+1; n,p).$$ Verify this (a) from the definition, (b) analytically.
$b(k; n, p)$ denote the binomial distribution of $k$ successes from $n$ trials. Although I am not sure if this helps, I found some recurrent relation $$B(k;n,p) - pb(k; n,p) = B(k-1; n,p) + (1-p)b(k;n,p)=\frac{n+1}{k}b(k-1; n,p)+B(k-2; n,p) +(1-p)b(k-1;n,p).$$
But, I am not sure how to go further. Could you give some hint please?
Let $X_{n}$ and $B$ be independent random variables defined on the same probability space.
This with $X_{n}\sim\mathsf{Bin}\left(n,p\right)$ and $B\sim\mathsf{Bernoulli}\left(p\right)$ and $p+q=1$.
Then $X_{n}+B\sim\mathsf{Bin}\left(n+1,p\right)$ and:
$\begin{aligned}P\left(X_{n}+B\leq k\right) & =P\left(X_{n}+B\leq k\mid B=0\right)P\left(B=0\right)+P\left(X_{n}+B\leq k\mid B=1\right)P\left(B=1\right)\\ & =P\left(X_{n}\leq k\mid B=0\right)q+P\left(X_{n}\leq k-1\mid B=1\right)p\\ & =P\left(X_{n}\leq k\right)q+P\left(X_{n}\leq k-1\right)p\\ & =P\left(X_{n}\leq k\right)q+\left[P\left(X_{n}\leq k\right)-P\left(X_{n}=k\right)\right]p\\ & =P\left(X_{n}\leq k\right)-pP\left(X_{n}=k\right) \end{aligned} $
For the second question observe that:
$$\left\{ X_{n}+B\leq k+1\right\} =\left\{ X_{n}\leq k\right\} \cup\left\{ B=0,X_{n}=k+1\right\}$$
This leads to: $$P\left(X_{n}+B\leq k+1\right)=P\left(X_{n}\leq k\right)+P\left(B=0,X_{n}=k+1\right)=$$$$P\left(X_{n}\leq k\right)+P\left(B=0\right)P\left(X_{n}=k+1\right)=P\left(X_{n}\leq k\right)+qP\left(X_{n}=k+1\right)$$