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If you have ever tried making patterns with a collection of coins, you have probably noticed that you can make hexagons in a natural way by packing circles as tightly as possible. The figure below shows how 19 circles fit into a hexagonal shape with 3 circles on each edge. Let H(n) be the number of circles you need to form a hexagon with n circles on each edge. From the figure below, it is clear that H(2) = 7 and H(3) = 19. It can be shown that increasing the number of circles on each edge gives the following recurrence relation: H(n) = 1 if n = 1 H(n − 1) + 6n − 6 if n > 1. Calculate H(9).

Respuesta :

barnuts

We are given the two equations:

H(n) = 1                                if n = 1                                                   --> eqtn 1

H(n) = H(n-1) + 6n – 6     if n>1                                     --> eqtn 2

 

Since we are to find for H(9), so obviously we would use eqtn 2. However take note that in eqtn 2, we have a factor which says H(n-1), this means that we have to calculate also for the value of H(8). However for H(8) we also have to calculate for H(7), therefore we need to calculate all values of H from 8 to 1 also. Let us start from 1:

 

H(1) = 1

H(2) = 1 + 6(2) – 6 = 7

H(3) = 7 + 6(3) – 6 = 19

H(4) = 19 + 6(4) – 6 = 37

H(5) = 37 + 6(5) – 6 = 61

H(6) = 61 + 6(6) – 6 = 91

H(7) = 91 + 6(7) – 6 = 127

H(8) = 127 + 6(8) – 6 = 169

H(9) = 169 + 6(9) – 6 = 217

 

Therefore the answer is:

H(9) = 217

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