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 4 of 4: Check the base case and conclude
Detailed analysis
For a one-digit positive number there are no stumps, so T(n)=0 and n=S(n). The induction step above now proves n=S(n)+9T(n) for every positive integer n.