Problem of induction Puns

Could someone briefly explain to me the relationship between empirical evidence and the problem of induction and also how falsification solves the problem of induction?

I’m very new to philosophy and am struggling to wrap my head around some of these ideas for university.

I am reading on the topic within Stephen Law "The Philosophy Gym" and I believe Popper tried to counter it with falisification, but that too has the same problem addressed in the chapter. Anyhow, help would be appreciated!

I would like to approach the study of Kant; particularly, one of the most important problems which he inherits. The following is what I've understood from the texts so far: I was hoping that people here could help read over and critique my view, which I am uncertain of. Apologies if this is an askphilosophy thread; I couldn't isolate a specific question so I decided to present all of it here. (In the spirit of transparency, this interpretation is heavily indebted to Deleuze--that's one of the reasons why I'm looking for another pair of eyes on this). Here it is:

Hume was the first philosopher to realize that most of our knowledge (namely, our knowledge of facts) is synthetic. The term is Kant’s, but Hume discovered its content and sense. What does it mean, for knowledge to be synthetic, knowing as a synthesis? Synthesis means going beyond what is given to us by our senses and memory. This is how Hume characterizes knowledge: I do not know when I say, ‘I have seen a thousand sunrises’ (because these are impressions preserved by the faculty of memory)—but I do know when I say, ‘The sun will rise tomorrow.’ This is because there is nothing given to me which tells me what will happen tomorrow; I therefore go beyond the given (senses and memory) when I make such a statement. This is the case with all inductive reasoning, the object of which is knowledge of facts about the world: when I say that water always boils at 100C (a fundamental law of chemistry), I go beyond the given, because neither my memory nor me senses can give me ‘always.’

Hume, within this epistemological framework, formulates a question in an extraordinary manner. He does not dispute the truth of knowledge, the truth of the statement, ‘The sun will rise tomorrow.’ In his eyes, this creates an unhealthy skepticism leading only to confusion and malaise. Nor does he ask, ‘How do we know?,’ because for Hume reasoning is simply an operation of the human mind, a part of human nature. Hume’s question is: What are the grounds for this going beyond? According to what right to we use inductive reasoning? Why do we believe in the knowledge which inductive reasoning gives us? This is the problem of induction.

Hume offers two stages to this question. The first is a critique of previous dogmas on this point, typical assumptions made about inductive knowledge. The next is a constructive solution to this question of grounds. Critically: Hume argues that we cannot rationally justify our belief in induct

n over sigma k = 1 (4k-1) = 2n^2 + n

Need to find assumption m-1 ( already proven to work )

Can anybody show me proof of "if k is odd, then 2^(n+2) divides k^(2n) - 1 for all natural numbers n."

I can't Google some problems so any advice on Induction problems (or number theory) book with solutions?

"Prove using mathematical induction that n(P(A)) = 2^(n(A))." P(A) being the power set of set A

Hey guys can someone help me with this problem? If you do a full solution pls post a pic

"Show that if the difference of 2 squares is divisible by 2, then their difference is divisible by 4 (use induction)"

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**Most detailed photo of over 265.000

Jeez I thought Briggs Myers went out with the ark. Can’t believe people still take notice of this unscientific horse shit.