In a sequence of numbers in which each term after the first term is 1 more than twice the preceding term, what is the fifth term? (1) The first term is 1. (2) The sixth term minus the fifth term is 32. DS21222
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Recurrence is a_{n+1} = 2a_n + 1. Statement (1) gives a_1 directly, easy. Statement (2) gives a_6 - a_5 = 32, but note a_6 = 2a_5 + 1, so a_6 - a_5 = a_5 + 1. That actually pins down a_5 without needing a_1. Took me a second to see that both work independently.
yeah the a_5 + 1 simplification is the whole trick, once you see it statement 2 is basically handing you the answer
same, I initially thought I needed the first term for statement 2 too
Is this really 505-555? The algebra isn't hard but recognizing that statement 2 is self-sufficient feels like a classic trap.
I think it's the trap that makes it feel harder, the actual math is pretty basic
Statement 1: a1=1, so the terms are 1, 3, 7, 15, 31... fifth term is clear. For statement 2, a6 = 2a5 + 1, so a6 - a5 = a5 + 1 = 32, meaning a5 = 31. Both give the same thing.
Classic DS mindset: you don't actually need to compute a value, just determine whether you CAN. Both statements give you enough.
this. spent way too long actually solving for the fifth term instead of just checking sufficiency
What it tests
Your ability to spot and extend arithmetic and geometric sequences and work with recursive definitions.
Common trap
Off-by-one errors in indexing (a₀ vs a₁) or confusing an arithmetic difference with a geometric ratio.