## Sum of 7’s

In the sum of the expression $7+77+777+7777+\cdots+7,\!777,\!777,\!777,\!777,\!777,\!777$, what digit will be in the tens place?
Source: NCTM Mathematics Teacher, February 2008

Solution
We are adding nineteen numbers from the smallest $7$ to the largest $7,\!777,\!777,\!777,\!777,\!777,\!777$. Note that there are nineteen $7\mathrm{'s}$ in the ones place and eighteen $7\mathrm{'s}$ in the tens place. First, we add the nineteen $7\mathrm{'s}$ in the ones place and get $19\times 7=133$. Consider the number $133$. The $3$ in the ones place of $133$ will be the ones place digit of the final sum. The $3$ in the tens place of $133$ means $30$ and if we regroup $30$ with the eighteen $70\mathrm{'s}$, we get $30+18\times 70=1290$. The digit in the tens place of the final sum is $9$.

Answer: $9$