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Age 16 to 18 Challenge Level:

This is a bit like a geometric sequence except that the elements are vectors, the 'common ratio' is vector ${\bf a}$ and the multiplication is vector products.

In the first part of the question where $|{\bf a}|< 1$ use the fact that $|\sin \theta | < 1$ to show that the magnitudes of the vectors in the sequence decrease.

In the second half of the question take ${\bf a}={\bf i}$ and ${\bf b_1} =r{\bf j}$