Permutations and Combinations

Permutations and Combinations

Question

An insect starts from a point and travels in a straight path 1 mm in the first second and half of the distance covered in the previous second in the succeeding second. In what time would it reach a point 3 mm away from its starting point?

Answer

Let a mm be the distance travelled in the first second. Since the insect covers half the distance covered in the previous second in the succeeding second.
∴   Distances covered in various seconds are 1 comma space 1 half comma space 1 fourth comma space 1 over 8........
Let r be the common ratio
∴                                    straight r space equals space 1 half
Let 3 mm distance be covered in n seconds
∴                  straight S subscript straight n space equals space 3
rightwards double arrow             fraction numerator straight a left parenthesis 1 minus straight r to the power of straight n right parenthesis over denominator 1 minus straight r end fraction space equals space 3 space space rightwards double arrow space space space fraction numerator 1 open square brackets 1 minus open parentheses begin display style 1 half end style close parentheses to the power of straight n close square brackets over denominator 1 minus begin display style 1 half end style end fraction space equals space 3 space rightwards double arrow space space space space 1 space minus space open parentheses 1 half close parentheses to the power of straight n space equals space 3 over 2 space rightwards double arrow space space open parentheses 1 half close parentheses to the power of straight n space equals space fraction numerator negative 1 over denominator 2 end fraction
which is impossible for any value of n.
Hence, the insect would never reach a point 3 mm  away from its starting point.

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