AMC 8 Daily Practice - The Sum of a Finite Arithmetic Series

Complete problem set with solutions and individual problem pages

Problem 4 Medium

A retirement community has 20 residents with integer ages summing to 1520. The oldest resident is aged >90 but \leq 100, while the remaining 19 residents have consecutive ages. What is the age range of the community?

  • A.

    19

  • B.

    29

  • C.

    30

  • D.

    66

  • E.

    95

Answer:B

Let minimum age be n. The sequence is: \underbrace{n, n+1, \dots, n+18}_{19 \text{ terms}}, \quad m

Sum equation: \frac{19}{2}[2n + 18] + m = 1520 \implies 19n + 171 + m = 1520 \implies m = 1349 - 19n

Given 90 < m \leq 100, solve: 90 < 1349 - 19n \leq 100 \implies 65.7 \leq n < 66.3 \implies n = 66

Thus m = 95, and age range = 95 - 66 = 29.

Final result: \boxed{29}

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