In my textbook practice problem, I want to prove the following by induction:
$$\sum_{i=0}^{n}\sum_{k=0}^{n-i}a_{i, k} = \sum_{m=0}^{n}\sum_{k=0}^{m}a_{m-k, k}$$
For my "$n+1$ implies $n$" statement I did the following:
\begin{align*} \sum_{i=0}^{n+1}\sum_{k=0}^{n+1-i} a_{i, k} &= \sum_{i=0}^{n+1}\left(\sum_{k=0}^{n-i}a_{i, k} + a_{i, n+1-i}\right) \\ &= \sum_{i=0}^{n}\left(\sum_{k=0}^{n-i}a_{i, k} + a_{i, n+1-i}\right) + \sum_{k=0}^{-1}a_{i, k} + a_{n+1, 0} \\ &= \sum_{i=0}^{n}\sum_{k=0}^{n-i}a_{i, k} + \sum_{i=0}^{n}a_{i, n+1-i} + a_{n+1, 0} \end{align*}
By the induction assumption,
$$ \sum_{i=0}^{n+1}\sum_{k=0}^{n+1-i} a_{i, k}= \sum_{m=0}^{n}\sum_{k=0}^{m}a_{m-k, k} + \sum_{i=0}^{n+1}a_{i, n+1-i}$$
Now for the last line, I see that there is no way to sum together the last two terms into
$$\sum_{m=0}^{n+1}\sum_{k=0}^{m}a_{m-k, k}$$
since the indices k and i do not match. I am not certain where I went wrong or if I'm misunderstanding something simple.
It's almost there. Indeed, \begin{align*} \sum_{i=0}^{n+1}\sum_{k=0}^{n+1-i} a_{i, k} &= \sum_{i=0}^{n+1}\left(\sum_{k=0}^{n-i}a_{i, k} + a_{i, n+1-i}\right) \\ &= \sum_{i=0}^{n}\left(\sum_{k=0}^{n-i}a_{i, k} + a_{i, n+1-i}\right) + \sum_{k=0}^{-1}a_{n+1, k} + a_{n+1, 0} \\ &= \sum_{i=0}^{n}\sum_{k=0}^{n-i}a_{i, k} + \sum_{i=0}^{n}a_{i, n+1-i} + a_{n+1, 0} \\ &= \sum_{m=0}^n \sum_{k=0}^m a_{m-k, k} + \sum_{i=0}^{n+1} a_{i,n+1-i}. \end{align*} Now, \begin{align*} \sum_{i=0}^{n+1} a_{i, n+1-i} &= a_{0, n +1} + a_{1, n} + a_{2, n-1} + \cdots + a_{n, 1} + a_{n+1, 0} \\ &= a_{n+1, 0} + a_{n,1} + \cdots + a_{1,n} + a_{0, n+1} \\ &= \sum_{k=0}^{n+1} a_{n+1-k, k}. \end{align*} So, \begin{align*} \sum_{i=0}^{n+1} \sum_{k=0}^{n+1-i} a_{i,k} &= \sum_{m=0}^n \sum_{k=0}^m a_{m-k, k} + \sum_{k=0}^{n+1} a_{n+1-k, k} \\ &= \sum_{m=0}^{n+1} \sum_{k=0}^m a_{m-k,k}, \end{align*} as required.