The sequence of four numbers , , , and is such that each number after the first is greater than the preceding number. What is the value of ? (1) (2)
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Wait, the common difference is a1 - 1? That's unusual, but I guess it means d depends on the first term. Took me a second to set up the equations.
Yeah same, I almost used a generic d and got stuck until I reread it.
Got a2 = a1 + (a1 - 1) = 2a1 - 1. Statement 1 gives 2a1 - 1 = 15, so that works. Statement 2 gives a4 = a1 + 3(a1 - 1) = 29, also solvable. So D?
That's what I got too. Nice that both give the same a1 actually.
Is this really 500-level? The a1-1 as the difference tripped me up. I kept thinking it was just a constant like usual.
Same here, the wording is the tricky part, not the math.
Quick note: since the difference contains a1 itself, you can't just treat it like a normal arithmetic sequence with a fixed d. Set up each term in terms of a1 and it becomes straightforward.
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.