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So I'm merrily going about my maths work then I get to this question and I'm like, the fuck? After about 2 hours I'm like, the fuck?
So as you can guess I have no idea about this. Any help would be great.

The question asks to prove by mathematical induction (n=1, n=k, n=k+1) that this formula holds for n > 0.
The formula being: Where L realates to Lucas numbers

L1 => 1 sum = 1
L2 => 3 sum = 4
L3 => 4 sum = 8
L4 => 7 sum = 15
L5 => 11 sum = 26
L6 => 18 sum = 44
L7 => 29 sum = 73
L8 => 47 sum = 120

I know how to do the actual induction, it's just in the formula you can't, as it needs the Lucas number. I tried using the Lucas number formula to do it, but got stuck when using k for the power.
Lucus number formula:

Can anyone help me, please?

I think you'll have better luck sending this message to your professor/teacher.

Don't you know how induction works?
I'm assuming you've done the base case.
Here you go.