MHB Question regarding pigeonhole method , number induction

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The discussion revolves around a user's struggle with the pigeonhole method and number induction, specifically in solving questions 4 and 5. The user presents a recurrence relation and derives a general solution, ultimately arriving at a formula for an. Forum members emphasize the importance of clarity in presenting mathematical work, suggesting that typing out equations would enhance understanding. Additionally, they note that the context for one of the problems is missing, which could hinder effective assistance. Overall, the thread highlights the need for clear communication and structured questions in mathematical discussions.
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Hi i have few questions over here and need some expert/pro view with approaching the question. The proving one would be an issue to me as well. Hope people would be able to assist me here , because i am basically struggling with it :( thanks

Questions :

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My Working :

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No idea how to approach Question 4 and 5
 

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Question 4

substitute rn for an
we get rn = 6rn-1 - 8rn-2
so r2 = 6r -8 or (r-4)(r-2)=0 so r = 4 or 2

A general solution is thus of the form an = A+ B

Now use a0 = 2 and a1 = 72 = A + B and 7 = 2A + 4B solving these gives = 0.5 and B = 1.5

The formula is thus an = 0.5(2n) + 1.5(4n)
 
Welcome to the forum, ffdude!

Please take some time to read the http://mathhelpboards.com/rules/. In particular, rule #8 recommends asking just one or at most two questions in one thread. Also, it would be much easier to understand your work if you typed it instead of taking a photo of it. It does not seem to have complicated formulas, so it can be typed even without the powerful formula editor on the right of the edit box. Finally, problem 1(a) refers to the description of an ice-cream palour, which is missing in the picture.
 
I'm taking a look at intuitionistic propositional logic (IPL). Basically it exclude Double Negation Elimination (DNE) from the set of axiom schemas replacing it with Ex falso quodlibet: ⊥ → p for any proposition p (including both atomic and composite propositions). In IPL, for instance, the Law of Excluded Middle (LEM) p ∨ ¬p is no longer a theorem. My question: aside from the logic formal perspective, is IPL supposed to model/address some specific "kind of world" ? Thanks.
I was reading a Bachelor thesis on Peano Arithmetic (PA). PA has the following axioms (not including the induction schema): $$\begin{align} & (A1) ~~~~ \forall x \neg (x + 1 = 0) \nonumber \\ & (A2) ~~~~ \forall xy (x + 1 =y + 1 \to x = y) \nonumber \\ & (A3) ~~~~ \forall x (x + 0 = x) \nonumber \\ & (A4) ~~~~ \forall xy (x + (y +1) = (x + y ) + 1) \nonumber \\ & (A5) ~~~~ \forall x (x \cdot 0 = 0) \nonumber \\ & (A6) ~~~~ \forall xy (x \cdot (y + 1) = (x \cdot y) + x) \nonumber...
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