2022 AMC 10B Problems/Problem 6
- The following problem is from both the 2022 AMC 10B #6 and 2022 AMC 12B #3, so both problems redirect to this page.
Contents
Problem
How many of the first ten numbers of the sequence are prime numbers?
Solution 1 (Generalization)
The th term of this sequence is It follows that the terms are Therefore, there are prime numbers in this sequence.
~MRENTHUSIASM
Solution 2 (Detailed Explanation of Solution 1)
Denote this sequence as , then we can find that So, we can induct that the general term is Therefore, there are prime numbers in this sequence.
~PythZhou
Solution 3 (Simple Sums)
Observe how all take the form of Factoring each of the sums, we have respectively. With each number factored, there are primes in the set.
~ab2024
Solution 4 (Educated Guess)
Note that is divisible by and is divisible by . Because this is Problem 6 of the AMC 10, we assume we do not need to check two-digit prime divisibility or use obscure theorems. Therefore, the answer is
~Dhillonr25
Video Solution(1-16)
~~Hayabusa1
Video Solution by Interstigation
https://youtu.be/_KNR0JV5rdI?t=562
Video Solution by Math4All999
https://youtu.be/5QYh3hNaDa0?feature=shared
See Also
2022 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
2022 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 2 |
Followed by Problem 4 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.