Prepswift Sequences 1: exercise question 8

GRE Quant Problem Solving
1

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I did not actually attempt this question because I was so overwhelmed by the first equation and did not know how to approach this in a timely way lol. I just guessed 2 (was coincidentally correct) and decided to skip to the solution - but I am confused about the solution now.

Firstly: to get a3 = 2 : am I expected to actually solve the equation? I couldnt find a time-efficient way to work it out and I was wondering if Greg would have a shortcut that I missed? He mentions the GRE calculator - I feel like putting all that in (and also worrying about rounding the root5) would somehow give me the wrong answer and I’d be stressed about messing something up. Alternatively, solving it by hand also took me a bit. (I expanded everything using binomial theory for ^3, and then simplified - this took me a long time)

Secondly: if I just get a1=1, a2=1, and a3=2, wouldn’t that be sufficient to find p+q = 2?

I’m confused as to why we have to also find a4=3 and then solve the system of equations? Since that would take me even longer. :smiling_face_with_tear:

2

Do you notice that the terms raised to the nth power are the golden ratio and its conjugate? If not, your method of expanding things out is also okay. You can make it faster by recognizing that you’re dealing with a difference of cubes:

a^3 - b^3 = (a - b)(a^2 + ab + b^2)

where a = 1 + \sqrt{5} and b = 1 - \sqrt{5}.

That should directly give you (1 + \sqrt 5)^3 - (1 - \sqrt 5)^3 = 16 \sqrt 5

Sure