I'd like to see a proof of this identity I discovered:
$$
\sum_{i=1}^{n}\sum_{j=1}^{i}a_j = \sum_{i=1}^{n}a_i(n-i+1)
$$
Probably there's something I forgot because I couldn't manage it.
[edit]Possibly with just identities, not examples please!
[edit2]No induction procedure, either.
Best Answer
$$\sum_{i=1}^n\sum_{j=1}^nx_{ij}=\sum_{i=1}^n\sum_{j=1}^nx_{ji}\;;\tag{1}$$ because each summand $x_{rs}$ on the RHS can be found on the LHS and vice versa and summation is commutativ. Therefore this sum is often written as
$$ \sum_{1\le i,j\le n}x_{ij}$$
We set
$$ x_{rs}=a_s,s \le r \;;\tag{2}$$ $$ x_{rs}=0, s \gt r \;;\tag{3}$$
Expanding the LHS from (1) and substituting (2) and (3) we get
$$\sum_{i=1}^n\sum_{j=1}^nx_{ij}=\sum_{i=1}^n\sum_{j=1}^{i}x_{ij}+\sum_{i=1}^n\sum_{j=i+1}^nx_{ij}=\sum_{i=1}^n\sum_{j=1}^{i}a_j+\sum_{i=1}^n\sum_{j=i+1}^n0=\sum_{i=1}^n\sum_{j=1}^ia_j$$
This is the LHS of your idendity.
Expanding the RHS of (1) and substituting (2) and (3) gives
$$\sum_{i=1}^n\sum_{j=1}^nx_{ji}=\sum_{i=1}^n\sum_{j=1}^{i-1}x_{ji}+\sum_{i=1}^n\sum_{j=i}^nx_{ji}=\sum_{i=1}^n\sum_{j=1}^{i-1}0+\sum_{i=1}^n\sum_{j=i}^na_i=\sum_{i=1}^na_i(\sum_{j=i}^n1)=\sum_{i=1}^na_i(\sum_{j=1}^n1-\sum_{j=1}^{i-1}1)=\sum_{i=1}^na_i(n-(i-1))$$
This is the RHS of your idendity.