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In the formula, xn{\displaystyle x_{n}} = the term in the sequence you are trying to find, n{\displaystyle n} = the position number of the term in the sequence, and {\displaystyle \phi } = the golden ratio. This is a closed formula, so you will be able to calculate a specific term in the sequence without calculating all the previous ones. This formula is a simplified formula derived from Binet's Fibonacci number formula. The formula utilizes the golden ratio ({\displaystyle \phi }), because the ratio of any two successive numbers in the Fibonacci sequence are very similar to the golden ratio. The n{\displaystyle n} represents whatever term you are looking for in the sequence. For example, if you are looking for the fifth number in the sequence, plug in 5. Your formula will now look like this: x5{\displaystyle x_{5}}=5(1)55{\displaystyle {
rac {\phi ^{5}-(1-\phi )^{5}}{\sqrt {5}}}}. You can use 1.618034 as an approximation of the golden ratio. For example, if you are looking for the fifth number in the sequence, the formula will now look like this: x5{\displaystyle x_{5}}=(1.618034)5(11.618034)55{\displaystyle {
rac {(1.618034)^{5}-(1-1.618034)^{5}}{\sqrt {5}}}}. Remember to use the order of operations by completing the calculation in parentheses first: 11.618034=0.618034{\displaystyle 1-1.618034=-0.618034}. In the example, the equation becomes x5{\displaystyle x_{5}}=(1.618034)5(0.618034)55{\displaystyle {
rac {(1.618034)^{5}-(-0.618034)^{5}}{\sqrt {5}}}}. Multiply the two parenthetical numbers in the numerator by the appropriate exponent. In the example, 1.6180345=11.090170{\displaystyle 1.618034^{5}=11.090170}; 0.6180345=0.090169{\displaystyle -0.618034^{5}=-0.090169}. So the equation becomes x5=11.090170(0.090169)5{\displaystyle x_{5}={
rac {11.090170-(-0.090169)}{\sqrt {5}}}}. Before you divide, you need to subtract the two numbers in the numerator. In the example, 11.090170(0.090169)=11.180339{\displaystyle 11.090170-(-0.090169)=11.180339}, so the equation becomes x5{\displaystyle x_{5}}=11.1803395{\displaystyle {
rac {11.180339}{\sqrt {5}}}}. The square root of 5, rounded, is 2.236067. In the example problem, 11.1803392.236067=5.000002{\displaystyle {
rac {11.180339}{2.236067}}=5.000002}. Your answer will be a decimal, but it will be very close to a whole number. This whole number represents the number in the Fibonacci sequence. If you used the complete golden ratio and did no rounding, you would get a whole number. It's more practical to round, however, which will result in a decimal. In the example, after using a calculator to complete all the calculations, your answer will be approximately 5.000002. Rounding to the nearest whole number, your answer, representing the fifth number in the Fibonacci sequence, is 5.
Set up the formula xn{\displaystyle x_{n}}=n(1)n5{\displaystyle {
rac {\phi ^{n}-(1-\phi )^{n}}{\sqrt {5}}}}. Plug the number for n{\displaystyle n} into the formula. Substitute the golden ratio into the formula. Complete the calculations in parentheses. Calculate the exponents. Complete the subtraction. Divide by the square root of 5. Round to the nearest whole number.