In 1202,Leonardo Fibonacci posed the following famous problem:Suppose a particul
ID: 2940004 • Letter: I
Question
In 1202,Leonardo Fibonacci posed the following famous problem:Suppose a particular breed of rabbits breed 1 new pair of rabbitseach month,except that a 1-month-old pair is too young tobreed.Suppose that no rabbits breeds with anyother except its paired mate and that rabbits live forever.At1 month we have our original pair of rabbits.At 2 months we stillhave the single pair.at 3 months,we have 2 pairs(the original andtheir 1 pair of offspring).At 4 months we have 3 pairs(the originalpair,1older pair of offspring,and one new pair of offspring). Find the formula for fn+3 -fn+1. In 1202,Leonardo Fibonacci posed the following famous problem:Suppose a particular breed of rabbits breed 1 new pair of rabbitseach month,except that a 1-month-old pair is too young tobreed.Suppose that no rabbits breeds with anyother except its paired mate and that rabbits live forever.At1 month we have our original pair of rabbits.At 2 months we stillhave the single pair.at 3 months,we have 2 pairs(the original andtheir 1 pair of offspring).At 4 months we have 3 pairs(the originalpair,1older pair of offspring,and one new pair of offspring). Find the formula for fn+3 -fn+1.Explanation / Answer
Consider the number of breeding pairs (label B) and the newly bornnon-breeding pairs (label N). The total number of pairs for each month is fn = Bn + Nn The number of breeding pairs will be the total number of pairs theprevious month Bn = fn-1 The number of new born pairs will be the number of breeding pairsthe previous month. So Nn = Bn-1 and from above, Bn-1 = fn-2 , soNn = fn-2 Putting it all together gives fn = Bn + Nn = fn-1 +fn-2 So if we use these in the expression given fn+3 - fn+1 = fn+2 +fn+1 - fn+1 = fn+2