The Fibonacci sequence must be familiar to anyone reading this. We define it by means of a second-order recurrence relation,

and two initial values, and . This immediately yields the well-known sequence

We can obtain an explicit expression for by solving the recurrence relation (1). Seeking a solution of the form , we find two possible values, and . We recognize as the** Golden Number**. The general solution of (1) is and, using the initial values, the complete solution is

This is known as **Binet’s Formula**, after the French mathematician Jacques Philippe Marie Binet (1786–1856), although it had been found by Abraham de Moivre a century earlier.

**Partial Sums of .**

The aim now is to find an expression for the partial sums of the sequence:

Clearly, the first few values are easily found:

By inspection, we see that is equal to , at least for the initial terms. We can prove this by means of the recurrence relation (1).

We write the sequence of recurrence relations

The sum of the left-hand terms is . On the right, all but two of the terms cancel, leaving and bringing us the required result,

**Is this expression of any value?**

Mathematics teachers and lecturers are all too familiar with the question *“But what use is this?” * In the present case, we may well ask if (3) is of practical use. It could be said that, to find , we need to know . But this requires additions, while direct evaluation of requires .

Of course, the worth of (3) lies in its generality. It is not a matter of calculating the terms , but in realising that the two sequences, and are so intimately related to each other. From the definition of we have . Combining this with (3) we get

We note that the recurrence relation (4) for is an inhomogeneous version of the corresponding relation (1) for . It has a particular solution . We thus seek a solution

and, using the initial conditions, we find and so that

This also follows directly from (2) and (3).

**Asymptotic Values**

Since and , the second terms in (2) and (5) become negligible for large , so that, for large ,

The nearest integer gives the exact value for ; the nearest integer minus 1 gives the exact value for . We may also note that, for large ,

So, and grow exponentially at the same rate, while is times bigger than .

**Proofs Without Words**

In a recent article in *The Mathematical Intelligencer* (Gryszka, 2022), a “proof without words” for the partial sums of the Fibonacci sequence was presented. It is shown in the figure below. You may find it interesting, although it took me longer to understand it than to find the simple algebraic proof presented above.

**Sources**

Karol Gryszka, 2022: Proof Without Words: Sum of Fibonacci Numbers and Beyond. *Math. Intelligencer*, **44**(4), 371–372. M. I. Article.