Change of index variable of summation (proof).

Question

Using this definition of finite series:

I'm trying to prove part (b):

It seems this should be easy to prove, but I can't seem to do it using just the definition and (a).

Answer 1

Since for all $i$, if we define $j:=i+k$, then $a_i=a_{j-k}$, then the sequence ${(a_i)}_{i=m}^n$ is identical to the sequence ${(a_{j-k})}_{j=m+k}^{n+k}$. So too must be the series be identical. $$\sum_{i=m}^n a_i=\sum_{j=m+k}^{n+k}a_{j-k}$$

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By induction: $${a_m=a_{(m+k)-k}\\~\\\forall n\geq m:\left[\sum_{i=m}^{n}a_i=\sum_{j=m+k}^{n+k}a_{j-k}~~\to~~a_{n+1}+\sum_{i=m}^{n}a_i=a_{(n+k)+1-k}+\sum_{j=m+k}^{n+k}a_{j-k}\right]}$$ So...

Answer 2

If $n<m$, then $n+k<m+k$ and so both sums are equal to $0$.

Otherwise, you can prove it by induction on $n-m$.