x_n = cα^n + dβ^n,
where α, β are the roots of the quadratic x²−ax−b; c, d are solutions to the system
c + d = x_0 cα + dβ = x_1.
If the series Σ x_n⋅10^n converges then its value is
10c/(10−α) + 10d/(10−β) = ((100−10a)x_0 + 10x_1)/(100 − 10a − b).
If a, b, x_0, x_1 are all rational then the above series converges to a rational number, too. This is the case for the Fibonacci sequence, with a=b=1, x_0=0 and x_1=1.
x_n = cα^n + dβ^n,
where α, β are the roots of the quadratic x²−ax−b; c, d are solutions to the system
c + d = x_0 cα + dβ = x_1.
If the series Σ x_n⋅10^n converges then its value is
10c/(10−α) + 10d/(10−β) = ((100−10a)x_0 + 10x_1)/(100 − 10a − b).
If a, b, x_0, x_1 are all rational then the above series converges to a rational number, too. This is the case for the Fibonacci sequence, with a=b=1, x_0=0 and x_1=1.