Difference between revisions of "2016 AIME II Problems/Problem 9"
(→Solution 1) |
|||
| Line 4: | Line 4: | ||
==Solution 1== | ==Solution 1== | ||
Since all the terms of the sequences are integers, and 100 isn't very big, we should just try out the possibilities for <math>b_2</math>. When we get to <math>b_2=9</math> and <math>a_2=91</math>, we have <math>a_4=271</math> and <math>b_4=729</math>, which works, therefore, the answer is <math>b_3+a_3=81+181=\boxed{262}</math>. | Since all the terms of the sequences are integers, and 100 isn't very big, we should just try out the possibilities for <math>b_2</math>. When we get to <math>b_2=9</math> and <math>a_2=91</math>, we have <math>a_4=271</math> and <math>b_4=729</math>, which works, therefore, the answer is <math>b_3+a_3=81+181=\boxed{262}</math>. | ||
| − | |||
| − | |||
== Solution 2== | == Solution 2== | ||
Revision as of 13:23, 20 June 2019
Contents
Problem
The sequences of positive integers 
and 
are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let 
. There is an integer 
such that 
and 
. Find 
.
Solution 1
Since all the terms of the sequences are integers, and 100 isn't very big, we should just try out the possibilities for 
. When we get to 
and 
, we have 
and 
, which works, therefore, the answer is 
.
Solution 2
Using the same reasoning (
isn't very big), we can guess which terms will work. The first case is 
, so we assume the second and fourth terms of 
are and 
. We let 
be the common ratio of the geometric sequence and write the arithmetic relationships in terms of .
The common difference is 
, and so we can equate: 
. Moving all the terms to one side and the constants to the other yields
, or 
. Simply listing out the factors of 
shows that the only factor 
less than a square that works is 
. Thus 
and we solve from there to get 
.
Solution by rocketscience
See also
| 2016 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 8 |
Followed by Problem 10 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
