Good 1974 reciprocal series fibonacci numbers

../

theorem_p346: The sum of the reciprocals of the Fibonacci numbers F_1, F_2, F_4, F_8, F_16 and onward along the powers of two equals (7 minus the square root of 5)/2, a quadratic irrational.

---

Good, I. J., A reciprocal series of Fibonacci numbers. Fibonacci Quart. 12 (1974), no. 4, 346.

A one-page note proving that the series 1/F_1 + 1/F_2 + 1/F_4 + 1/F_8 + 1/F_16 + ... equals (7 - sqrt 5)/2. The proof is the finite identity 1/F_1 + 1/F_2 + ... + 1/F_{2^n} = 3 - F_{2^n - 1}/F_{2^n}, which the paper says is proved by induction with Binet's formula (it does not state the range of n; the identity holds for every n ≥ 1), followed by letting n tend to infinity. Good remarks that the result resembles a formula for sqrt(m), m > 1, from his earlier note with T. N. Gover (his Reference 1), built from the quadratically recurrent sequence a_{n+1} = a_n^2 - 2, and quotes a divisibility curiosity about 5^{F_n} from his earlier joint work with R. A. Gaskins (his Reference 2). For Erdős problem 267, which asks whether the sum of 1/F_{n_k} must be irrational whenever n_{k+1}/n_k ≥ c > 1, the theorem is the instance n_k = 2^{k-1} (ratio exactly 2), whose value is an explicit quadratic irrational; it answers the question only for this one sequence.

Source: https://www.fq.math.ca/12-4.html. No notice is printed on the one-page scan; the journal's volume contents page shows the site-wide footer "Copyright © 2010 The Fibonacci Association. All rights reserved." and names no license (https://www.fq.math.ca/12-4.html, read 2026-10-02), every other right reserved.

Results.

• Theorem (p. 346, unnumbered): 1/F_1 + 1/F_2 + 1/F_4 + 1/F_8 + ... = (7 - sqrt 5)/2, the indices running over the powers 2^k, k ≥ 0; the page also records the finite identity behind the proof.

Read status. Claims checked: the Theorem and the identity in its proof were read on p. 346 of the printed note.

Bears on. #267 (the Theorem evaluates the sum for the single index sequence n_k = 2^{k-1}, ratio exactly 2, as the quadratic irrational (7 - sqrt 5)/2; the paper does not mention the general question or any other sequence)

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.