Proving a summation using mathematical induction and partial fractions

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Discrete Mathematics
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Question 1
For n≥ 1 : F1 + F3+ F5 +... ... ... ... F2 n−1=∑
k=1
n
F2k −1=F2 n
Prove by Induction
Let the induction basis is as shown below:
n=1
F1=1
F2=1
Now,
F2∗1−1=F2∗1
It means the statement holds for n=1
Now,
Inductive step
Let said that the statement holds for n = m and thus,
F1+ F3 +... ... ... ...+ F2 m−1=F2 m
For n=m+ 1 ,
F1+ F3 +... ... ... ... + F2(m+1 )−1=
F1+ F3 +... ... ... ...+ F2 m+ 2−1=¿
F1+ F3 +... ... ... ...+ F2 m+1 =¿
Now,
1