12345
Repeat a pattern of numbers to form a larger number. Can you find the sum of all the digits?
Age
14 to 16
Challenge level
Problem
The pattern 123451234512345... is continued to form a 2000 digit number.
What is the sum of all 2000 digits?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Answer: $6000$
$\underbrace{\overbrace{12345}^{15}\overbrace{12345}^{15}\overbrace{12345}^{15}\dots\overbrace{12345}^{15}}_\text{2000 digits}$
Total: $15\times(2000\div5) = 6000$
Alternatively, the mean of each group of five digits is $3$ and so the mean of the digits making up the number is $3$. Therefore the sum is $2000\times 3=6000$.
Note that this only works because the groups are in $5$s, and $5$ goes into $2000$ exactly