Converging Product

In the limit you get the sum of an infinite geometric series. What about an infinite product (1+x)(1+x^2)(1+x^4)... ?

A circle is inscribed in an equilateral triangle. Smaller circles touch it and the sides of the triangle, the process continuing indefinitely. What is the sum of the areas of all the circles?

Binary Squares

If a number N is expressed in binary by using only 'ones,' what can you say about its square (in binary)?

Clickety Click and All the Sixes

Why do this problem?

A non-standard problem based on place value and summing geometric series requiring simple manipulation of a numerical expression.

Possible approach

A short problem suitable as a lesson starter.

Key questions

Can you write the 10 digit number $6666666666$ as a geometric series?

Can you sum this series?

Can you do the same for any number that is a 'string of sixes'?

Possible extension

For a discussion of a generalisation of this problem see Sum The Series.