Problem 1
For a positive integer n, let S(n) be the sum of the digits in the decimal representation of n. A positive integer obtained by removing at least one digit from the right-hand end of n is called a stump of n. Let T(n) be the sum of all stumps of n. Prove that n=S(n)+9T(n).
Step 3 of 4: Propagate the induction identity
Detailed analysis
By the induction hypothesis n-S(n)=9T(n). Therefore m-S(m)=9T(n)+9n=9(T(n)+n)=9T(m). This proves the identity for a number with one more digit whenever it holds for n.