# Sum of Sequence of Fibonacci Numbers

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## Theorem

Let $F_n$ denote the $n$th Fibonacci number.

Then:

- $\ds \forall n \in \Z_{\ge 0}: \sum_{j \mathop = 0}^n F_j = F_{n + 2} - 1$

## Proof

Proof by induction:

For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:

- $\ds \sum_{j \mathop = 0}^n F_j = F_{n + 2} - 1$

$\map P 0$ is the case:

\(\ds F_0\) | \(=\) | \(\ds 0\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 1 - 1\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds F_2 - 1\) |

which is seen to hold.

### Basis for the Induction

$\map P 1$ is the case:

\(\ds F_1\) | \(=\) | \(\ds 1\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2 - 1\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds F_3 - 1\) |

which is seen to hold.

This is our basis for the induction.

### Induction Hypothesis

Now we need to show that, if $\map P k$ is true, where $k \ge 2$, then it logically follows that $\map P {k + 1}$ is true.

So this is our induction hypothesis:

- $\ds \sum_{j \mathop = 1}^k F_j = F_{k + 2} - 1$

Then we need to show:

- $\ds \sum_{j \mathop = 1}^{k + 1} F_j = F_{k + 3} - 1$

### Induction Step

This is our induction step:

\(\ds \sum_{j \mathop = 1}^{k + 1} F_j\) | \(=\) | \(\ds \sum_{j \mathop = 1}^k F_j + F_{k + 1}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds F_{k + 2} - 1 + F_{k + 1}\) | Induction Hypothesis | |||||||||||

\(\ds \) | \(=\) | \(\ds F_{k + 3} - 1\) | Definition of Fibonacci Number |

So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.

Therefore:

- $\ds \forall n \in \Z_{\ge 0}: \sum_{j \mathop = 0}^n F_j = F_{n + 2} - 1$

$\blacksquare$

## Also presented as

This can also be seen presented as:

- $\ds \sum_{j \mathop = 1}^n F_j = F_{n + 2} - 1$

which is seen to be equivalent to the result given, as $F_0 = 0$.

## Sources

- 1971: George E. Andrews:
*Number Theory*... (previous) ... (next): $\text {1-1}$ Principle of Mathematical Induction: Exercise $7$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $5$ - 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.8$: Fibonacci Numbers: Exercise $20$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $5$