Line generation punch card 01

As with the high numbers, it is a good idea to put some proofs.
$Look at a simple proof question today:$
$If A ^ 2 = A, verify: (A + E) ^ k = E + (2 ^ k-1) A.$
$certificate$
$Prove with mathematical induction. When k = 1, the conclusion holds, and the inductive hypothesis holds true for k = m. That is, (A + E) ^ m = E + (2 ^ m-1) A. When k = m + 1 ， 有 (A + E) ^ {m + 1} = [E + (2 ^ m-1) A] (A + E) = A + (2 ^ m-1) A ^ 2 + E + (2 ^ m-1 ) A = A + (2 ^ m-1) A + E + (2 ^ m-1) A = [2 (2 ^ m-1) +1] A + E = E + (2 ^ {m + 1} -1 ) A.$