How Can You Express Fibonacci Terms f_{n+1} and f_{n} Using f_{n+2} and f_{n+3}?

[morbello]

i have to find the recurrence relation to express both f$$_{n+1}$$ and f$$_{n}$$ with f$$_{n+2}$$ and f$$_{n+3}$$

my answer is
f$$_{n+1}$$+f$$_{n+2}$$+f$$_{n+3}$$...f$$_{n}$$

ive got other ways off doing it in the book but they do not use f$$_{n+3}$$

these are

f$$_{n+2}$$+f$$_{f-1}$$-f$$_{n}$$


they all should be in lower case but i can not seam to get the post to do it i hope it does not make it hard to help me with the question.

The Attempt at a Solution

 

[lanedance]

hi morbello

you can click on any latex code to see how it is typed, easiest to put tex quotes around your whole equation


but I'm not too sure what you're trying to do... ;) shouldn't you have an equals sign somewhere?

something like
$$f^{n+1} = f(n+1) = f(n) + ...?$$

 

[Architeuthis Dux]

:

The Fibonacci recurrence relation is a mathematical formula that describes the relationship between consecutive terms in the Fibonacci sequence. This sequence is defined as a series of numbers where each number is the sum of the two preceding numbers, starting with 0 and 1. The recurrence relation for this sequence is f_{n+2} = f_{n+1} + f_{n}.

To express both f_{n+1} and f_{n} with f_{n+2} and f_{n+3}, we can use the relation f_{n+3} = f_{n+2} + f_{n+1}. This means that f_{n+1} = f_{n+3} - f_{n+2} and f_{n} = f_{n+2} - f_{n+1}. Substituting these expressions into the original recurrence relation, we get f_{n+2} = (f_{n+3} - f_{n+2}) + (f_{n+2} - f_{n+1}). Simplifying this equation, we get f_{n+3} = 2f_{n+2} - f_{n+1}.

Therefore, the recurrence relation to express both f_{n+1} and f_{n} with f_{n+2} and f_{n+3} is f_{n+3} = 2f_{n+2} - f_{n+1}. This relation can be used to find any term in the Fibonacci sequence by plugging in the appropriate values for n.