Question

. Neve is examining a tree limb. She notices that the smaller branches spiral away from the larger limb. The limb has 1 branch, then another single branch. After that, the branches spiral in the pattern {2, 3, 5, 8, …}. Can you model the tree growth with the Fibonacci sequence? Explain.

Answer 1

Yes.

Explanation:

The pattern in which the branches are growing off the limb is {1, 1, 2, 3, 5, 8, …}, meaning you can use the model with tree growth.

Answer 2

`a_1=2\\a_n=a_(n-1)+(n-1)\,,n>1`

Explanation:

Given: The smaller branches spiral in the pattern
`\{2,3,5,8,...\}` from a larger limb.

To model: the tree growth with the Fibonacci sequence

Solution:

Take
`a_1=2`

Let
`a_n` denotes the
`n^(th)` term.

`a_2=3=2+1=2+(2-1)=a_1+(2-1)\\a_3=5=3+2=3+(3-1)=a_2+(3-1)\\a_4=8=5+3=5+(4-1)=a_3+(4-1)`

So,

`a_1=2\\a_n=a_(n-1)+(n-1)\,,n>1`