Question

The Fibonacci sequence $1,1,2,3,5,8$ occurs in nature. What are the ninth and tenth terms in the Fibonacci sequence? Is the Fibonacci sequence arithmetic, geometric, both, or either?

Answer 1

Hint: The Fibonacci sequence is the sum of two preceding numbers. The Fibonacci series are never ending series which last for infinite numbers.
The arithmetic mean is the Sequence of terms which is the addition of terms and subtraction of constant terms.
The geometric mean is the number of terms meant to be multiplied or exponential.

Complete step-by-step answer:
Given,
The sequence is $1,1,2,3,5,8$ .
To find the nth terms in Fibonacci sequence,
${t_n} = {t_{n - 1}} + {t_{n - 2}}$
From the given series
$
  {t_1} = 1 \\
  {t_2} = 1 \\
  {t_3} = 2 \\
  {t_4} = 3 \\
  {t_5} = 5 \\
  {t_6} = 8 \\
 $
We need to find ${t_9},{t_{10}}$ .
Substitute $n = 9$in ${t_n} = {t_{n - 1}} + {t_{n - 2}}$
${t_9} = {t_8} + {t_7}$
We need to first find out ${t_7}$and ${t_8}$
Calculate ${t_7} = {t_6} + {t_5}$
Substitute ${t_5} = 5$ and ${t_6} = 8$
${t_7} = 5 + 8$
Add ${t_7} = 13$
Calculate ${t_8} = {t_7} + {t_6}$
Substitute ${t_7} = 13$ and ${t_6} = 8$
${t_8} = 13 + 8$
Add ${t_8} = 21$
Calculate ${t_9} = {t_8} + {t_7}$
Substitute ${t_7} = 13$ and ${t_8} = 21$
${t_9} = 13 + 21$
Add ${t_9} = 34$
Calculate ${t_{10}} = {t_9} + {t_8}$
Substitute ${t_9} = 34$ and ${t_8} = 21$
${t_{10}} = 34 + 21$
Add ${t_{10}} = 55$
The $9$ th and $10$ th terms are $34$ and $55$ respectively.
To check whether arithmetic or geometric, it must satisfy certain equations.
For arithmetic ${t_2} = \dfrac{{{t_1} + {t_3}}}{2}$
Arithmetic mean is the mean of the terms which are in arithmetic sequence.
From given substitute
$
  {t_1} = 1 \\
  {t_2} = 1 \\
  {t_3} = 2 \\
 $
$
  {t_2} = \dfrac{{{t_1} + {t_3}}}{2} \\
  {t_2} = \dfrac{{2 + 1}}{2} \\
  {t_2} \ne \dfrac{3}{2} \\
 $
By comparing the values in the right side and left side of the equation, hence confirm that the Fibonacci sequence is not arithmetic.
For geometric ${t_2} = \sqrt {{t_1} \times {t_3}} $
From given substitute
$
  {t_1} = 1 \\
  {t_2} = 1 \\
  {t_3} = 2 \\
 $
$
  {t_2} = \sqrt {{t_1} \times {t_3}} \\
  {t_2} = \sqrt {1 \times 2} \\
  {t_2} \ne \sqrt 2 \\
 $
By comparing the values on the right side and left side of the equation. We confirm that the Fibonacci sequence is not geometric.
By substituting the values from Fibonacci series in arithmetic mean and geometric mean formula we confirm that Fibonacci series are neither arithmetic mean nor geometric mean.
The $9$ th and $10$ th terms are $34$ and $55$ respectively and the Fibonacci sequence is neither geometric or arithmetic.

Note: The nth terms formula must be correct. There must be correct substitution of values. The arithmetic and geometric mean formula must be correct. Before and last terms should be calculated correctly. If one term is valued wrongly, then the whole answer would be wrong. Always remember the formulae for the arithmetic and geometric mean.