Problem 11
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The sequence $1, x, y, z$ is arithmetic. The sequence $1, p, q, z$ is geometric. Both sequences are strictly increasing and contain only integers, and $z$ is as small as possible. What is the value of $x+y+z+p+q$?
$\textbf{(A) } 66 \qquad\textbf{(B) } 91 \qquad\textbf{(C) } 103 \qquad\textbf{(D) } 132 \qquad\textbf{(E) } 149$
Difficulty: 23/100 — Fairly straightforward and approachable to beginners. Solvable by trial and error.
Core Concepts: arithmetic & geometric sequences, trial and error
Challenges: Figuring out a systematic way to find smallest $z$.
Techniques: Trial and error to find $z$.
Error-prone Steps: Finding an incorrect value of $z$ or adding incorrectly.
Ideal Time:
Experienced: ≤ 45 sec
Intermediate: ≤ 1.5 min
Beginner: ≤ 3-4 min