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Question

A line initially 1 inch long grows according to the following law, where the first term is the initial length. 1+142+14+1162+116+1642+164+ If the growth process continues forever, the limit of the length of the line is:

Accepted Answer

We can rewrite our sum as the sum of two infinite geometric sequences. 1 + 142 + 14 + 1162 + 116 + ... = (1 + 14 + 116 + 164 + ...) + (142 + 1162 + 1642 + ...) (1 + 14 + 116 + 164 + ...) + 2(14 + 116 + 164 + ...) We now take the sum of each of the infinite geometric sequences separately (1 + 14 + 116 + 164 + ...) + 2(14 + 116 + 164 + ...) = 11 - 14 + 2(141 - 14) 43 + 23 13(4 + 2) Therefore, the answer is 13(4+2)