MHB Question regarding pigeonhole method , number induction

[ffdude]

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

 

[IanCg]

Question 4

substitute rⁿ for a_n
we get rⁿ = 6r^n-1 - 8r^n-2​
so r² = 6r -8 or (r-4)(r-2)=0 so r = 4 or 2

A general solution is thus of the form a_n = 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 a_n = 0.5(2ⁿ) + 1.5(4ⁿ)

 

[Evgeny.Makarov]

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.