Prove that $F(1) + F(3) + F(5) + ... + F(2n-1) = F(2n)$

Question

(These are Fibonacci numbers; $f(1) = 0$, $f(3) = 1$, $f(5) = 5$, etc.) I'm having trouble proving this with induction, I know how to prove the base case and present the induction hypothesis but I'm unfamiliar with proving series such as this. Any help would be great. :)

Answer 1

Assume this is true for $ n $. Now, let's consider the sum:

$$ F_1 + F_3 + \dots + F_{2n-1} + F_{2n+1}$$

By induction hypothesis, the sum without the last piece is equal to $ F_{2n} $ and therefore it's all equal to:

$$ F_{2n} + F_{2n+1} $$

And it's the definition of $ F_{2n+2} $, so we proved that our induction hypothesis implies the equality:

$$ F_1 + F_3 + \dots + F_{2n-1} + F_{2n+1} = F_{2n+2}$$

Which finishes the proof

Answer 2

Hint: If you can use the that the sequence of Fibonacci numbers is defined by the recurrence relation $$F(n)=F(n-1)+F(n-2)$$ then you can prove it by induction, since $$\begin{align*}F(2(n+1))&=F(2n+2)\\&=F(2n+1)+F(2n)\\&=F(2n+1)+\underbrace{F(2n-1)+\ldots+F(5)+F(3)+F(1)}_{=F(2n) \text{ by induction hypothesis}}\end{align*}$$

Answer 3

The following is formally not correct because it uses the "$\cdots$" symbol but it gives some insight. The proof can be formalized using induction. This was already done by @Jytug. $$\begin{eqnarray} F(1) + F(3) + F(5) + \cdots + F(2n-1)&= \\ 0+\left(F(1) + F(3) + F(5) + \cdots + F(2n-1)\right) &= \\ F(0)+\left(F(1) + F(3) + F(5) + \cdots + F(2n-1)\right)&=\\ \left(F(0)+F(1)\right) +\left( F(3) + F(5) + \cdots + F(2n-1)\right)&=\\ F(2)+\left( F(3) + F(5) + \cdots + F(2n-1)\right)&=\\ \left(F(2)+F(3)\right) + \left( F(5) + \cdots + F(2n-1)\right)&=\\ F(4) + \left( F(5) + \cdots + F(2n-1)\right)&=\\ \cdots&=\\ F(2n-2)+F(2n-1)&=\\ F(2n)& \end{eqnarray} $$

Answer 4

Here is a solution using Binet's formula:

$F(n) = a \alpha^n + b \beta^n$, for some $a,b \in \mathbb R$, and $\alpha,\beta$ the roots of $X^2=X+1$. (Their precise value is not really important here.)

Now, $ \alpha^1 + \alpha^3 + \cdots + \alpha^{2n-1} =\alpha(1+\alpha^2 + \alpha^4 + \cdots + \alpha^{2n-2}) =\alpha\dfrac{\alpha^{2n}-1}{\alpha^2-1} =\alpha^{2n}-1 $ because $\alpha=\alpha^2-1$.

Analogously, $ \beta^1 + \beta ^3 + \cdots + \beta ^{2n-1} =\beta ^{2n}-1 $.

Therefore, $ F(1) + F(3) + F(5) + \cdots + F(2n-1) = a(\alpha^{2n}-1)+b(\beta ^{2n}-1)=F(2n)-a-b $.

Finally, $a+b=F(0)=0$.

Answer 5

Since $F(2n)-F(2n-2)=F(2n-1)$, this reduces to a telescoping series: $$ \begin{align} \sum_{k=1}^nF(2k-1) &=\sum_{k=1}^n(F(2k)-F(2k-2))\\ &=\sum_{k=1}^nF(2k)-\sum_{k=0}^{n-1}F(2k)\\ &=F(2n)-F(0)\\[9pt] &=F(2n) \end{align} $$