A man arranges to pay off a debt of $3600 by 40 annual installments which are in arithmetic progression. When 30 of the installments are paid he dies leaving one-third of the debt unpaid. What was the value of the 8th installment?
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I got thrown off because there are two APs here: the installments and the cumulative sum. Once I set the first installment as a and common difference as d, the 40th term and sum conditions gave me two equations. Solving them was fine after that. Answer is 65.
Wait, how did you get the second equation? I used the fact that the sum of the first 30 installments is 2/3 of the total debt.
This one is tricky because the wording makes it seem like the installments themselves are one-third unpaid, but it's actually the cumulative sum after 30 payments. Once I fixed that, it was straightforward. Got C.
Is there a quicker way to do this? I ended up solving the two equations by elimination, but it took me a while. Maybe plugging in answer choices for the 8th term?
You could test choices, but setting up the equations is probably faster if you're comfortable with algebra. The numbers are nice.
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.