2011 AMC 10A Problem 17

Below is the professionally curated solution for Problem 17 of the 2011 AMC 10A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2011 AMC 10A solutions, or check the answer key.

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Concepts:recursion pattern recognition

Difficulty rating: 1370

17.

In the eight term sequence $A,$ $B,$ $C,$ $D,$ $E,$ $F,$ $G,$ $H,$ the value of $C$ is $5$ and the sum of any three consecutive terms is $30.$ What is $A+H?$

$17$

$18$

$25$

$26$

$43$

Solution:

From the condition about the sequence, we get that $A + B + C = 30$ $B = 25 - A.$ Similarly, we get $B + C + D = 30$ $D = A.$

Propagating these values through the sequence and repeating the condition for every consecutive triple, we get that $E = 25 - A, F = 5, G = A,$ and finally, $H = 25 - A.$

The desired sum is then $A + 25 - A = 25.$

Thus, C is the correct answer.

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