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Difference between revisions of "2022 AMC 10B Problems/Problem 6"

(Created page with "==Problem== How many of the first ten numbers of the sequence <math>121, 11211, 1112111, \ldots</math> are prime numbers? <math>\textbf{(A) } 0 \qquad \textbf{(B) }1 \qquad \...")
 
Line 5: Line 5:
  
 
==Solution==
 
==Solution==
−
The recursive formula for this sequence is
+
The <math>n</math>th term of this sequence is
 
<cmath>\begin{align*}
 
<cmath>\begin{align*}
−
a_1 &= 121, \\
+
\left(10^{2n}+10^{2n-1}+\cdots+10^{n}\right)+\left(10^{n}+10^{n-1}+\cdots+10^{0}\right) &= 10^n\left(10^{n}+10^{n-1}+\cdots+10^{0}\right)+\left(10^{n}+10^{n-1}+\cdots+10^{0}\right) \\
−
a_n &= 10a_{n-1} + 10^{2n} + 1 & \text{for }2\leq n\leq 10.
+
&= \left(10^n + 1\right)\left(10^{n}+10^{n-1}+\cdots+10^{0}\right).
 
\end{align*}</cmath>
 
\end{align*}</cmath>
 +
It follows that the first ten terms are divisible by <math>11,101,1001,\ldots,</math> respectively.
 +
 +
Therefore, there are \boxed{\textbf{(A) } 0} prime numbers in this sequence.
 +
 
~MRENTHUSIASM
 
~MRENTHUSIASM
  

Revision as of 15:30, 17 November 2022

Problem

How many of the first ten numbers of the sequence $121, 11211, 1112111, \ldots$ are prime numbers?

$\textbf{(A) } 0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3 \qquad \textbf{(E) }4$

Solution

The $n$th term of this sequence is \begin{align*} \left(10^{2n}+10^{2n-1}+\cdots+10^{n}\right)+\left(10^{n}+10^{n-1}+\cdots+10^{0}\right) &= 10^n\left(10^{n}+10^{n-1}+\cdots+10^{0}\right)+\left(10^{n}+10^{n-1}+\cdots+10^{0}\right) \\ &= \left(10^n + 1\right)\left(10^{n}+10^{n-1}+\cdots+10^{0}\right). \end{align*} It follows that the first ten terms are divisible by $11,101,1001,\ldots,$ respectively.

Therefore, there are \boxed{\textbf{(A) } 0} prime numbers in this sequence.

~MRENTHUSIASM

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

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