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\textit{}\begin{problem}  \textbf{Problem1: \textbf{Problem 1:  }(a) You are eagerly waiting for the delivery of your smart phone. For each day the delivery is late, the phone company compensates you by giving you coupons. The number of coupons you get  for each day is equal to the number of days the delivery is late. For example, if your phone is  late $2$ days, you get $1$ coupon for the first day and $2$ coupons for the second day. 

\textbf{Solution 1: }  \begin{problem}  Prove that there is an integer $c>0$ such that  the following inequality holds for all $n\ge c$: