2023 AMC 8

Complete problem set with solutions and individual problem pages

Problem 22 Hard

In a sequence of positive integers, each term after the second is the product of the previous two terms. The sixth term is 4000. What is the first term?

  • A.

    1

  • B.

    2

  • C.

    4

  • D.

    5

  • E.

    10

Answer:D

Solution 1

In this solution, we will use trial and error to solve. 4000 can be expressed as 200 \times 20. We divide 200 by 20 and get 10, divide 20 by 10 and get 2, and divide 10 by 2 to get \boxed{\textbf{(D)}\ 5}. No one said that they have to be in ascending order!

 

Solution 2

Consider the first term is a and the second term is b. Then, the following term will be ab, ab^2, a^2b^3 and a^3b^5. Notice that 4000=2^5\times 5^3, then we obtain a=\boxed{\textbf{(D)}\ 5} and b=2.

Related Problems
Problem 20 2023 AMC 8
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