Fibonacci Sum

Calculus Level 5

n=1nFn2n=kn=1Fn2n \sum_{n=1}^ \infty \dfrac{ n F_{n}}{ 2^n } = k{\sum_{n=1}^ \infty \dfrac{ F_{n}}{ 2^n }}

Let FnF_n denote the nthn^\text{th} Fibonacci number, where F0=0F_0 = 0, F1=1F_1 = 1 and Fn=Fn1+Fn2F_n = F_{n-1} + F_{n-2} for n=2,3,4,....n=2,3,4, ....

Find the value of kk satisfying the equation above.

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