I'm trying prove the following statement but I'm having some trouble: $$\sum_{j=0}^{n}(2j-1) = F_{2n}$$
But I've come across some contradictions (which are likely due to my own fault). I've done the base base at n = 1, which checks out.
My inductive hypothesis is:
$$\sum_{j=0}^{k}(2j-1) = F_{2k}$$
And my inductive step is:
$$\sum_{j=0}^{k+1}(2j-1) = F_{2k+1}$$
From here I can get $$\sum_{j=0}^{k}(2j-1) + F_{2k+1}= F_{2k+1}$$
We can already see the problem before I substitute in the formula from the I.H.
Where have I gone wrong? Thanks for the help!
Induction allows us to prove some claim for all the naturals ($n\in\mathbb{N}$).
The claim is as follows:
$${\sum_{j=0}^{n}F_{(2j-1)}=F_{2n}}$$
Consider the base case, that is when $n=1$
$${\sum_{j=0}^{1}F_1= 1 + 1 =2} \ \checkmark$$
(Assuming you define the "0"th term as the element $1$ in the Fibonacci sequence.)
Suppose the $n^{th}$ case holds, such that:
$${\sum_{j=0}^{n}F_{(2j-1)}=F_{2n}}$$
Then we want to show the $n^{th+1}$ case holds, that is:
$${\sum_{j=0}^{n+1}F_{(2j-1)}=F_{2(n+1)}=F_{2n+2}}$$
We show this as follows:
$${\sum_{j=0}^{n+1}F_{(2j-1)}={\sum_{j=0}^{n}F_{(2j-1)}\ +F_{2n+1}=F_{2n}+F_{2n+1}=F_{2n+2}}}$$
Note, the Fibonacci sequence is obtained by simply adding the $n^{th+1}$ term to the $n^{th}$ term, producing the $n^{th+2}$ term. Also note in the $n^{th+1}$ case, the $n^{th+1}$ term is extracted from the summation, allowing us to substitute the original assumption. ($n^{th}$ case)