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Difference between revisions of "2023 AMC 10B Problems/Problem 6"

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<math>\textbf{(A) }673\qquad\textbf{(B)} 674\qquad\textbf{(C) }675\qquad\textbf{(D) }1010\qquad\textbf{(E) }1011</math>
 
<math>\textbf{(A) }673\qquad\textbf{(B)} 674\qquad\textbf{(C) }675\qquad\textbf{(D) }1010\qquad\textbf{(E) }1011</math>
  
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==Solution==
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==Solution 1==
  
 
We calculate more terms:
 
We calculate more terms:
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~Mintylemon66
 
~Mintylemon66
 +
 +
==Solution 2==
 +
Like in the other solution, we find a pattern, except in a more rigorous way.
 +
Since we start with <math>1</math> and <math>3</math>, the next term is <math>4</math>.
 +
 +
We start with odd, then odd, then (the sum of odd and odd) even, (the sum of odd and even) odd, and so on. Basically the pattern goes: odd, odd, even, odd odd, even, odd, odd even…
 +
 +
When we take <math> \frac{2023}{3}</math> we get <math>674</math> with a remainder of one. So we have <math>674</math> full cycles, and an extra odd at the end.
 +
 +
Therefore, there are <math>\boxed{\textbf{(B) }674}</math> evens.

Revision as of 16:56, 15 November 2023

Problem

Let $L_{1}=1, L_{2}=3$, and $L_{n+2}=L_{n+1}+L_{n}$ for $n\geq 1$. How many terms in the sequence $L_{1}, L_{2}, L_{3}......L_{2023}$ are even?

$\textbf{(A) }673\qquad\textbf{(B)} 674\qquad\textbf{(C) }675\qquad\textbf{(D) }1010\qquad\textbf{(E) }1011$

Solution 1

We calculate more terms:

$1,3,4,5,9,14,...$

We find a pattern: if $n$ is a multiple of $3$, then the term is even, or else it is odd. There are $\lfloor \frac{2023}{3} \rfloor =\boxed{\textbf{(B) }674}$ multiples of $3$ from $1$ to $2023$.

~Mintylemon66

Solution 2

Like in the other solution, we find a pattern, except in a more rigorous way. Since we start with $1$ and $3$, the next term is $4$.

We start with odd, then odd, then (the sum of odd and odd) even, (the sum of odd and even) odd, and so on. Basically the pattern goes: odd, odd, even, odd odd, even, odd, odd even…

When we take $\frac{2023}{3}$ we get $674$ with a remainder of one. So we have $674$ full cycles, and an extra odd at the end.

Therefore, there are $\boxed{\textbf{(B) }674}$ evens.

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