2009 AMC 12B Problem 12

Below is the professionally curated solution for Problem 12 of the 2009 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2009 AMC 12B solutions, or check the answer key.

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Concepts:geometric sequencefactorial

Difficulty rating: 1500

12.

The fifth and eighth terms of a geometric sequence of real numbers are 7!7! and 8!8! respectively. What is the first term?

6060

7575

120120

225225

315315

Solution:

The eighth term divided by the fifth term is r3=8!7!=8,r^3 = \dfrac{8!}{7!} = 8, so r=2.r = 2.

The fifth term is ar4=7!,a r^4 = 7!, so a=7!16=504016=315.a = \dfrac{7!}{16} = \dfrac{5040}{16} = 315.

Thus, the correct answer is E.

Problem 12 in Other Years