## Abstract

In this paper, we investigate the reciprocal sums of even and odd terms in the Fibonacci sequence, and we obtain four interesting families of identities which give the partial finite sums of the even-indexed (resp., odd-indexed) reciprocal Fibonacci numbers and the even-indexed (resp., odd-indexed) squared reciprocal Fibonacci numbers.

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## 1 Introduction

The *Fibonacci sequence* is defined by the linear recurrence relation

where \(F_{n}\) is called the *n*th *Fibonacci number* with \(F_{0}=0\) and \(F_{1}=1\). There exists a simple and non-obvious formula for the Fibonacci numbers,

The Fibonacci sequence plays an important role in the theory and applications of mathematics, and its various properties have been investigated by many authors; see [1–5].

In recent years, there has been an increasing interest in studying the reciprocal sums of the Fibonacci numbers. For example, Elsner *et al.* [6–9] investigated the algebraic relations for reciprocal sums of the Fibonacci numbers. In [10], the partial infinite sums of the reciprocal Fibonacci numbers were studied by Ohtsuka and Nakamura. They established the following results, where \(\lfloor\cdot\rfloor\) denotes the floor function.

### Theorem 1.1

*For all*
\(n\geq2\),

### Theorem 1.2

*For each*
\(n\geq1\),

Wu and Zhang [11, 12] generalized these identities to the Fibonacci polynomials and Lucas polynomials, and they considered the subseries of infinite sums derived from the reciprocals of the Fibonacci polynomials and Lucas polynomials.

Recently, Wu and Wang [13] studied the partial finite sum of the reciprocal Fibonacci numbers and deduced the following main result.

### Theorem 1.3

*For all*
\(n\geq4\),

Inspired by Wu and Wang’s work, Wang and Wen [14] strengthened Theorem 1.1 and 1.2 to the finite sum case.

### Theorem 1.4

*If*
\(m\geq3\)
*and*
\(n\geq2\), *then*

### Theorem 1.5

*For all*
\(m\geq2\)
*and*
\(n\geq1\), *we have*

Applying elementary methods, we investigate the partial finite sums of the even-indexed and odd-indexed reciprocal Fibonacci numbers in this paper, and obtain four interesting families of identities. In Section 2, we consider the reciprocal sums of even and odd terms in the Fibonacci sequence. In Section 3, we present the finite sums of the even-indexed and odd-indexed squared reciprocal Fibonacci numbers.

## 2 Main results I: the reciprocal sums

We first present several well-known results on Fibonacci numbers, which will be used throughout the article. The detailed proofs can be found in [5].

### Lemma 2.1

*Let*
\(n\geq1\), *we have*

*and*

*if*
*a*
*and*
*b*
*are positive integers*.

As a consequence of (2.2), we have the following result.

### Corollary 2.2

*If*
\(n\geq1\), *then*

The following is an interesting identity concerning the Fibonacci numbers.

### Lemma 2.3

*Assume that*
*a*
*and*
*b*
*are two integers with*
\(a\geq b\geq0\). *If*
\(n>a\), *then*

### Proof

We proceed by induction on *n*. It is clearly true for \(n=a+1\). Assuming the result holds for any integer \(n>a\), we show that the same is true for \(n+1\).

Applying (2.2) repeatedly and by the induction hypothesis, we get

which completes the induction on *n*. □

### Remark

Recently, Akyiğit *et al.* [15, 16] defined the split Fibonacci quaternion, the split Lucas quaternion and the split generalized Fibonacci quaternion, and they obtained some similar identities to those above for these quaternions.

Before presenting our main results, we establish an inequality.

### Proposition 2.4

*If*
\(n\geq3\), *then*

### Proof

A direct calculation shows that it is true for \(n=3\). Thus, we assume that \(n\geq4\) in the rest of the proof.

Setting \(a=2\) and \(b=0\), and replacing *n* by 2*n* in (2.6) yields

From (2.5), we know that

Applying (2.8), (2.9), and the fact \(F_{2n-3}\geq2\) and \(F_{2n-1}F_{2n}>F_{2n+1}\) if \(n\geq3\), we obtain

which is equivalent to

Now we have

It is not hard to see that for \(n\geq4\), \(F_{2n-3}\geq n+1\), which completes the proof. □

Now we introduce our main results on the reciprocal sums of Fibonacci numbers.

### Theorem 2.5

*For all*
\(n\geq3\), *we have*

### Proof

By elementary manipulations and (2.1), we derive that, for \(k\geq1\),

Hence, we have

It follows from (2.4) that

which implies that

Invoking (2.1) again, we can readily deduce that

from which we obtain

Because of (2.7), we get, if \(n\geq3\),

Combining (2.11) and (2.13), we have

which yields the desired identity. □

### Theorem 2.6

*If*
\(m\geq3\)
*and*
\(n\geq1\), *we have*

### Proof

It is obviously true for \(n=1\). Now we assume that \(n\geq2\).

By some calculations and (2.1), we obtain, for \(k\geq2\),

from which we have

On the other hand, it follows from (2.12) that

We claim that if \(n\geq1\) and \(m\geq3\),

Replacing *a* by \(a-1\) in (2.2), we arrive at

which implies that

Thus, \(F_{2n-1}F_{2n}F_{2n+1}\leq F_{6n}< F_{6n+1}\leq F_{2mn+1}\), which means

Combining (2.16) and (2.18) yields

from which the desired result follows immediately. □

### Corollary 2.7

*For all*
\(n\geq1\), *we have*

### Proof

By using (2.15) repeatedly, we have

Thus, we obtain

Applying the same argument to (2.12) yields

Hence we have

which completes the proof. □

### Remark

Identity (2.19) can be regarded as the limit of (2.14) as \(m\to\infty\).

### Theorem 2.8

*For all*
\(n\geq1\)
*and*
\(m\geq2\), *we have*

### Proof

It is clearly true for \(n=1\), hence we suppose that \(n\geq 2\) in the following.

Invoking (2.1), we derive that for \(k\geq2\),

which implies that

It follows from (2.17) that

based on which we conclude that, when \(n>1\),

Employing (2.1) again, we can readily obtain

from which we arrive at

Combining the above inequality with (2.21), we have

which yields the desired result. □

As *m* approaches infinity, Theorem 2.8 becomes the following.

### Corollary 2.9

*If*
\(n\geq1\), *we have*

## 3 Main results II: the reciprocal square sums

We first introduce several preliminary results on the square of the Fibonacci numbers.

### Lemma 3.1

*For all*
\(n\geq2\), *we have*

### Proof

It follows from

that

where the last equality follows from (2.1). □

### Lemma 3.2

*If*
\(n\geq2\), *then*

### Proof

It is straightforward to check that

where the last equality follows from (2.6). □

### Lemma 3.3

*For each*
\(n\geq2\), *we have*

### Proof

A direct calculation shows that

The proof is complete. □

### Remark

In fact, applying the equalities (ii) and (iv) of Proposition 2.2 of [17], we can easily obtain

where \(L_{n}\) means the *n*th Lucas number. Then (3.3) follows immediately from the fact \(L_{n}>F_{n}\) for \(n\geq2\).

Now we are ready to present the reciprocal square sums of the Fibonacci numbers.

### Theorem 3.4

*For all*
\(n\geq1\)
*and*
\(m\geq2\), *we have*

### Proof

It is clearly true for \(n=1\), so we assume that \(n\geq2\) in the rest of the proof.

For \(k\geq2\), we have

It follows from (2.3) that

As a consequence of (3.1), we see

Applying (2.1), (3.5), (3.6), and (3.7), we derive that

which implies that

Thus, we have

Employing the same argument as above, we obtain, for \(k\geq2\),

For each \(k\geq2\), we have

Therefore,

from which we arrive at

Combining (3.8) and (3.9) yields

from which the desired result follows. □

As *m* tends to infinity in Theorem 3.4, we have the following consequence.

### Corollary 3.5

*For all*
\(n\geq1\), *we have*

### Theorem 3.6

*If*
\(n\geq1\)
*and*
\(m\geq2\), *then*

### Proof

It is obvious when \(n=1\), thus we assume that \(n\geq2\) in the following.

It follows from (2.3) that

Therefore, applying (3.2), we deduce

For \(k\geq2\), we have

from which we derive

Employing (3.2) and (3.12), we obtain

where the last inequality follows from (3.3).

Now we see that, for \(k\geq2\),

which implies that

It is easy to see that

Hence,

It follows from (3.13) and (3.14) that

which completes the proof. □

Consequently, we have the following result.

### Corollary 3.7

*If*
\(n\geq1\), *then*

## 4 Conclusions

In this paper, we give the exact integral values of the reciprocal sums (resp., square sums) of the even and odd terms in the Fibonacci sequence. The results are new and important for those with closely related research interests. In addition, the methods used here are very elementary and can be extended to the investigation of other combinatorial sequences.

In a future paper, the reciprocal sums and the reciprocal square sums of the Fibonacci 3-subsequences will be presented.

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## Acknowledgements

The authors would like to thank the anonymous referees for their helpful suggestions and comments which improved significantly the presentation of the paper. This work was supported by the National Natural Science Foundation of China (No. 11401080).

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All authors contributed equally to deriving all the results of this article, and read and approved the final manuscript.

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Wang, A.Y., Zhang, F. The reciprocal sums of even and odd terms in the Fibonacci sequence.
*J Inequal Appl* **2015**, 376 (2015). https://doi.org/10.1186/s13660-015-0902-2

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DOI: https://doi.org/10.1186/s13660-015-0902-2