Polynomial of Fibonacci

Algebra Level 3

Let $P_n(x)$ be a polynomial of $n$ terms such that:

$P_n(x) = 1+x+2x^2+3x^3+5x^4+...+F_{n}x^{n-1}$

where $F_{n}$ is the $n^{\text{th}}$ Fibonacci number. Find the value of $\lim\limits_{n \to \infty}P_{n}\left(\frac{1}{2}\right)$ .

The answer is 4.

1 solution

Rohan Shinde
Mar 16, 2019

As I knew the intention to form the question (Using generating function of Fibonacci numbers) ,so it was meaningless to write a solution,but later thought that the solution might help future readers so here Is just the Generating function for Fibonacci numbers wherein you need to substitute $x=\frac 12$ $\displaystyle \sum_{i=1}^{\infty} F_i x^{i-1}=\frac {1}{1-x-x^2}$

Yup.....Although it would be better for the future readers if you provide a proof in your solution, or simply link a proof .....:)

Aaghaz Mahajan - 2 years, 2 months ago

Polynomial of Fibonacci

The answer is 4.

1 solution

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