2019 AMC 10A Problems/Problem 15

The following problem is from both the 2019 AMC 10A #15 and 2019 AMC 12A #9, so both problems redirect to this page.

Problem

A sequence of numbers is defined recursively by $a_1 = 1$ , $a_2 = \frac{3}{7}$ , and $a_n=\frac{a_{n-2} \cdot a_{n-1}}{2a_{n-2} - a_{n-1}}$ for all $n \geq 3$ Then $a_{2019}$ can be written as $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. What is $p+q ?$

$\textbf{(A) } 2020 \qquad\textbf{(B) } 4039 \qquad\textbf{(C) } 6057 \qquad\textbf{(D) } 6061 \qquad\textbf{(E) } 8078$

Solution 1 (Induction)

Using the recursive formula, we find $a_3=\frac{3}{11}$ , $a_4=\frac{3}{15}$ , and so on. It appears that $a_n=\frac{3}{4n-1}$ , for all $n$ . Setting $n=2019$ , we find $a_{2019}=\frac{3}{8075}$ , so the answer is $\boxed{\textbf{(E) }8078}$ .

To prove this formula, we use induction. We are given that $a_1=1$ and $a_2=\frac{3}{7}$ , which satisfy our formula. Now assume the formula holds true for all $n\le m$ for some positive integer $m$ . By our assumption, $a_{m-1}=\frac{3}{4m-5}$ and $a_m=\frac{3}{4m-1}$ . Using the recursive formula, $a_{m+1}=\frac{a_{m-1}\cdot a_m}{2a_{m-1}-a_m}=\frac{\frac{3}{4m-5}\cdot\frac{3}{4m-1}}{2\cdot\frac{3}{4m-5}-\frac{3}{4m-1}}=\frac{\left(\frac{3}{4m-5}\cdot\frac{3}{4m-1}\right)(4m-5)(4m-1)}{\left(2\cdot\frac{3}{4m-5}-\frac{3}{4m-1}\right)(4m-5)(4m-1)}=\frac{9}{6(4m-1)-3(4m-5)}=\frac{3}{4(m+1)-1},$ so our induction is complete.

Solution 2

We have $\frac{1}{a_n} = \frac{2a_{n-2}-a_{n-1}}{a_{n-2} \cdot a_{n-1}}=\frac{2}{a_{n-1}}-\frac{1}{a_{n-2}}$ , in other words, $\frac{1}{a_n}-\frac{1}{a_{n-1}} = \frac{1}{a_{n-1}}-\frac{1}{a_{n-2}}$ . So $\{\frac{1}{a_n}\}$ is an arithmetic sequence with step size $\frac{7}{3}-1=\frac{4}{3}$ , which means $\frac{1}{a_{2019}} = 1+2018 \cdot \frac{4}{3} = \frac{8075}{3}$ . Since the numerator and the denominator are relatively prime, the answer is $\boxed{\textbf{(E) } 8078}$ .

-eric2020 (modified by Dolphindesigner)

Solution 3

It seems reasonable to transform the equation into something else. Let $a_{n}=x$ , $a_{n-1}=y$ , and $a_{n-2}=z$ . Therefore, we have $x=\frac{zy}{2z-y}$ $2xz-xy=zy$ $2xz=y(x+z)$ $y=\frac{2xz}{x+z}$ Thus, $y$ is the harmonic mean of $x$ and $z$ . This implies $a_{n}$ is a harmonic sequence or equivalently $b_{n}=\frac{1}{a_{n}}$ is arithmetic. Now, we have $b_{1}=1$ , $b_{2}=\frac{7}{3}$ , $b_{3}=\frac{11}{3}$ , and so on. Since the common difference is $\frac{4}{3}$ , we can express $b_{n}$ explicitly as $b_{n}=\frac{4}{3}(n-1)+1$ . This gives $b_{2019}=\frac{4}{3}(2019-1)+1=\frac{8075}{3}$ which implies $a_{2019}=\frac{3}{8075}=\frac{p}{q}$ . $p+q=\boxed{\textbf{(E) } 8078}$ ~jakeg314

Solution 4 (Arithmetic Sequence)

Notice that $a_n = \frac{1}{\frac{2}{a_{n-1}} - \frac{1}{a_{n-2}}}.$ Therefore, $\frac{1}{a_n} = \frac{2}{a_{n-1}} - \frac{1}{a_{n-2}}, \ \ \implies \ \ \frac{\frac{1}{a_n} + \frac{1}{a_{n-2}}}{2} = \frac{1}{a_{n-1}}.$ Therefore, the sequence $b_n = \frac{1}{a_n}$ is an arithmetic sequence. Notice that the common difference of $b$ is $\frac{4}{3},$ and therefore $b_{2019} = b_1 + 2018 \bigg(\frac{4}{3}\bigg) = 1 + 2018 \bigg(\frac{4}{3} \bigg) = \frac{8075}{3}.$ Therefore, we see that $a_{2019} = \frac{3}{8075},$ so that $p + q = \boxed{\text{(E) } 8078}.$

~Professor-Mom

Note: This is similar to solutions #2 and #3, although you can notice that in #2's case the new sequence $B$ actually forms an arithmetic sequence.

Solution 5 (Characteristic Equation - Overkill but Generic)

We have $\frac{1}{a_n} = \frac{2a_{n-2}-a_{n-1}}{a_{n-2} \cdot a_{n-1}}=\frac{2}{a_{n-1}}-\frac{1}{a_{n-2}}$ ,

let $b_n = \frac{1}{a_n}$ , then $b_n = 2b_{n-1} - b_{n-2}$

, this is 2nd order linear homogeneous recurrence sequence,

the characteristic equation for this is $x^2 -2x +1 = 0$ , which has double root x=1

so $b_n = (c_1 + c_2 \cdot n) \cdot 1^n$

plug in $b_1 = \frac{1}{a_1} = 1 = c_1 + c_2 \cdot 1, b_2 = \frac{1}{a_2}= \frac{7}{3} =c_1 + c_2 \cdot 2$

we solve $c_1 = -\frac{1}{3} , c_2=\frac{4}{3}$ , so $b_n = -\frac{1}{3} + \frac{4}{3} \cdot n$

so ${b_{2019}} = -\frac{1}{3} + \frac{4}{3} \cdot 2019 = \frac{8075}{3}$ .

$a_{2019} = \frac{1}{b_{2019}} = \frac{3}{8075}$

Since the numerator and the denominator are relatively prime, the answer is $\boxed{\textbf{(E) } 8078}$ .

note: characteristic equation is overkill for this simple one but is more generic solution for other parameter values.

~luckuso

2019 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 14	Followed by Problem 16
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All AMC 10 Problems and Solutions

2019 AMC 12A (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 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

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