My grandmother started walking a mile a day when she was ninety.

We have no idea where she is now!

`That was just a joke`

If instead of walking a mile everyday, if she walked 99% of the distance she walked the previous day, at least we would know how far she would eventually have travelled assuming she walked one mile on the first day.

How far would she have travelled after a suitably large number of days?

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Consider the infinite(?) number sequence:

1, 0.99, 0.9801 ...

This is a geometric sequence with first term 1 and common ratio 0.9

The sum of the sequence is (1 - 0.99^{n})/(1 - 0.99)

When n is sufficiently large the term 0.99^{n} tends to zero.

The sum is then 1/(1 - 0.99)

= 1/0.01

= 100 miles!

Check this answer by generating the sequence and its sum on a spreadsheet

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