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∑n=1∞nFn2n=k∑n=1∞Fn2n \sum_{n=1}^ \infty \dfrac{ n F_{n}}{ 2^n } = k{\sum_{n=1}^ \infty \dfrac{ F_{n}}{ 2^n }}n=1∑∞2nnFn=kn=1∑∞2nFn
Let FnF_nFn denote the nthn^\text{th} nth Fibonacci number, where F0=0F_0 = 0F0=0, F1=1F_1 = 1F1=1 and Fn=Fn−1+Fn−2F_n = F_{n-1} + F_{n-2} Fn=Fn−1+Fn−2 for n=2,3,4,....n=2,3,4, ....n=2,3,4,....
Find the value of kkk satisfying the equation above.
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