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Imagine a $600$ cm snake that coils into a circle, but he's not going to eat himself.
He is going to coil himself around to make a double circle, then a treble circle and so on until he is coiled round like this rope.

Challenge 

Once in a circle how much further will he have to slither to make a double circle of his body?  What's this length as a fraction of his total length?

How much further will he have to slither to make three circles of his body?
What's this length as a fraction of his total length?

 

How much further will he have to slither to make four circles of his body?
What's this length as a fraction of his total length?

How much further will he have to slither to make five circles of his body?
What's this further length as a fraction of his total length?


Final challenge


When the snake has slithered a total of $550$cm along his body length, making smaller and smaller circles of his body, how many circles will there be?