Sum of Sequence of Odd Index Fibonacci Numbers
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Theorem
Let $F_k$ be the $k$th Fibonacci number.
Then:
\(\ds \forall n \ge 1: \, \) | \(\ds \sum_{j \mathop = 1}^n F_{2 j - 1}\) | \(=\) | \(\ds F_1 + F_3 + F_5 + \cdots + F_{2 n - 1}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds F_{2 n}\) |
Proof
Proof by induction:
For all $n \in \N_{>0}$, let $\map P n$ be the proposition:
- $\ds \sum_{j \mathop = 1}^n F_{2 j - 1} = F_{2 n}$
Basis for the Induction
$\map P 1$ is the case $F_1 = 1 = F_2$, which holds from the definition of Fibonacci numbers.
This is our basis for the induction.
Induction Hypothesis
Now we need to show that, if $\map P k$ is true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.
So this is our induction hypothesis:
- $\ds \sum_{j \mathop = 1}^k F_{2 j - 1} = F_{2 k}$
Then we need to show:
- $\ds \sum_{j \mathop = 1}^{k + 1} F_{2 j - 1} = F_{2 k + 2}$
Induction Step
This is our induction step:
\(\ds \sum_{j \mathop = 1}^{k + 1} F_{2 j - 1}\) | \(=\) | \(\ds \sum_{j \mathop = 1}^k F_{2 j - 1} + F_{2 k + 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds F_{2 k} + F_{2 k + 1}\) | Induction Hypothesis | |||||||||||
\(\ds \) | \(=\) | \(\ds F_{2 k + 2}\) | Definition of Fibonacci Numbers |
So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.
Therefore:
- $\ds \forall n \ge 1: \sum_{j \mathop = 1}^n F_{2 j - 1} = F_{2 n}$
$\blacksquare$
Sources
- 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {1-1}$ Principle of Mathematical Induction: Exercise $8$
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $5$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $5$