Consider the following series:\(1,2,3,4,5,10,20,40,\ldots,\)which starts as an arithmetic series, but after the first 5 terms becomes a geometric series. Prove that any positive integer can be written as a sum of distinct numbers from this series.

Find an expression for the sum of the \(i-\)th row of the following triangle, and prove the correctness of your claim. Each entry in the triangle is the sum of three entries directly above it(a nonexisting entry is considered 0).
\(\matrix{&&&&1&&&&\cr &&&1&1&1&&&\cr&&1&2&3&2&1&&\cr&1&3&6&7&6&3&1&\cr 1&4&10&16&19&16&10&4&1}\)

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