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

Answer:

$a_1=2\\a_n=a_{n-1}+(n-1)\,,n>1$

Step-by-step 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,

$a_1=2\\a_n=a_{n-1}+(n-1)\,,n>1$