The Fibonacci sequence of numbers begins 1,1,2,3,5,8,13,21,34,55,…and continues by the rule: every number is the sum of the two previous numbers.

a. Determine the next five (eleventh through fifteenth) elements of this sequence.
b. Compute the ratio of the fourteenth and fifteenth elements of this sequence. Is it close to the golden ratio?

1,1,2,3,5,8,13,21,34,55,89,144,233,377,610, ...

ratio of 15th/14th = 610/377 = 1.618037

the golden ratio = (1 + √5)/2 = 1.618033989..

for a difference of .0000031
(not bad)

a. Sure, here are the next five elements of the Fibonacci sequence:

11th element: 89
12th element: 144
13th element: 233
14th element: 377
15th element: 610

b. To compute the ratio of the 14th and 15th elements, we have:

Ratio = 377 / 610 ≈ 0.61803278688

This value is indeed close to the golden ratio, which is approximately 1.61803398875. It's like they are distant relatives, human and humarian if you will.

a. To find the next five elements, we can follow the rule that every number is the sum of the two previous numbers in the Fibonacci sequence.

11th element = 8 + 13 = 21
12th element = 13 + 21 = 34
13th element = 21 + 34 = 55
14th element = 34 + 55 = 89
15th element = 55 + 89 = 144

So the next five elements of the Fibonacci sequence are 21, 34, 55, 89, and 144.

b. To compute the ratio of the 14th and 15th elements, we divide the 14th element by the 15th element.

Ratio = 89 / 144 = 0.618055555556

The ratio is approximately 0.618, which is close to the golden ratio. The golden ratio is often approximated as 1.618.

To determine the next five elements of the Fibonacci sequence, you need to apply the rule that every number is the sum of the two previous numbers.

a. Here are the next five elements of the Fibonacci sequence:

11th element: 34 (sum of 21 and 13)
12th element: 55 (sum of 34 and 21)
13th element: 89 (sum of 55 and 34)
14th element: 144 (sum of 89 and 55)
15th element: 233 (sum of 144 and 89)

So, the eleventh through fifteenth elements of the Fibonacci sequence are 34, 55, 89, 144, and 233.

b. To compute the ratio of the fourteenth and fifteenth elements of the Fibonacci sequence, divide the 14th element (144) by the 15th element (233):

Ratio = 144 / 233 ≈ 0.618

The ratio of the fourteenth and fifteenth elements of the Fibonacci sequence is approximately 0.618. This ratio is very close to the golden ratio, which is approximately 1.618.