Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1977 AHSME Problems/Problem 13"

(Created page with "== Problem 13 == If <math>a_1,a_2,a_3,\dots</math> is a sequence of positive numbers such that <math>a_{n+2}=a_na_{n+1}</math> for all positive integers <math>n</math>, then...")
(No difference)

Revision as of 12:15, 21 November 2016

Problem 13

If $a_1,a_2,a_3,\dots$ is a sequence of positive numbers such that $a_{n+2}=a_na_{n+1}$ for all positive integers $n$, then the sequence $a_1,a_2,a_3,\dots$ is a geometric progression

$\textbf{(A) }\text{for all positive values of }a_1\text{ and }a_2\qquad\\ \textbf{(B) }\text{if and only if }a_1=a_2\qquad\\ \textbf{(C) }\text{if and only if }a_1=1\qquad\\ \textbf{(D) }\text{if and only if }a_2=1\qquad\\ \textbf{(E) }\text{if and only if }a_1=a_2=1$


Solution

Solution by e_power_pi_times_i

The first few terms are $a_1,a_2,a_1a_2,a_1a_2^2,a_1^2a_2^3,\dots$. If this is a geometric progression, $\dfrac{a_2}{a_1} = a_1 = a_2 = a_1a_2$. $a_1=0,1$, $a_2=0,1$. Since this is a sequence of positive integers, then the answer must be $\boxed{\textbf{(E) }\text{if and only if }a_1=a_2=1 }$

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1977_AHSME_Problems/Problem_13&oldid=81213"