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3 Responses
Yep-the “−3” happens at every step: a1=5, so a2=2·5−3=7, a3=2·7−3=11, and a4=2·11−3=19.
Analogy: each round you double your coins and then pay a 3-coin tax-every round, not just the first.
Think of the rule a_{n+1} = 2a_n − 3 like a little machine you run every time you make a new term: take the current term, double it, then subtract 3-rinse and repeat for each step, not just once. Starting at a1 = 5: a2 = 2·5 − 3 = 7, a3 = 2·7 − 3 = 11, and a4 = 2·11 − 3 = 19. So yes, the “−3” is sprinkled into every batch, not only the first. If you like a tidy formula, this kind of rule has the closed form a_n = 3 + (a_1 − 3)·2^{n−1}, which for a1 = 5 becomes a_n = 3 + 2^n, giving a4 = 3 + 16 = 19 again. Once you see it as “apply the same recipe each time,” the worry about when to subtract melts away. Hope this helps!
Subtract 3 at every step: starting a1=5 gives a2=2·5−3=7, then a3=2·7−3=11, and a4=2·11−3=19. It’s like seasoning each layer of lasagna-double the noodles, then shake off 3 grains every layer, not just the first.