Why do this problem?

An excellent exercise in using geometric series with a fun way into the problem through experimenting with numbers and making conjectures.

Possible approach

Let the students experiment with the numbers until they spot patterns and make conjectures. If they don't see how to prove the general result you may have to ask the 'key question' and suggest using the sum of a geometric series.

Key questions

Can you write the number (k sixes followed by a 7) in terms of powers of 10 which will work for all k?

Possible extension

Try the problem Clickety Click and All the Sixes