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Difference between revisions of "2016 AIME II Problems/Problem 9"

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==Problem==
 
The sequences of positive integers <math>1,a_2, a_3,...</math> and <math>1,b_2, b_3,...</math> are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let <math>c_n=a_n+b_n</math>. There is an integer <math>k</math> such that <math>c_{k-1}=100</math> and <math>c_{k+1}=1000</math>. Find <math>c_k</math>.
 
The sequences of positive integers <math>1,a_2, a_3,...</math> and <math>1,b_2, b_3,...</math> are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let <math>c_n=a_n+b_n</math>. There is an integer <math>k</math> such that <math>c_{k-1}=100</math> and <math>c_{k+1}=1000</math>. Find <math>c_k</math>.
  
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==Solution==
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==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>.
  
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Solution by Shaddoll
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== Solution 2 (Some trial and error)==
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We have <math>a_k=r^{k-1}</math> and <math>b_k=(k-1)d</math>. First, <math>b_{k-1}<c_{k-1}=100</math> implies <math>d<100</math>, so <math>b_{k+1}<300</math>.
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It follows that <math>a_{k+1}=1000-b_{k+1}>700</math>, i.e., <cmath>700 < r^k < 1000.</cmath> Moreover, since <math>k</math> is atleast <math>3</math> we get <math>r^3\le r^k <1000</math>, i.e. <math>r<10</math>. For every value of <math>r</math> in this range, define <math>i(r) = \max \{x : r^x < 700\}</math>, and define <math>j(r) = \min \{x : r^x > 1000\}</math>. We are looking for values of <math>r</math> such that <math>j(r) -i(r)>1</math>. Let's make a table:
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<cmath>\begin{array}[b]{ c c c c c c c c c }
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r & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\[2ex]
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i(r) & 9  & 5 & 4  & 4 & 3 & 3 & 3 & 2\\[2ex]
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j(r) & 10  & 7 & 5  & 5 & 4 & 4 & 4 & 4
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\end{array} </cmath>
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The only admissible values for <math>r^k</math> are <math>\{3^6, 9^3\}</math>. However, since <math>100=c_{k-1}=r^{k-2}+(k-2)d+1</math>, we must have <math>(k-2)\mid 99-r^{k-2}</math>. This does not hold for <math>r^k=3^6</math> because <math>4</math> does not divide <math>99-3^4=18</math>. This leaves <math>r^k=9^3</math> as the only option.
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For <math>r=9</math> and <math>k=3</math>, we check: <math>a_{k-1}= a_2 = r =9</math> implies <math>b_{k-1}= b_2 = 91</math>, i.e. <math>d=90</math>. Then <math>a_{k+1}=a_4 = r^3 = 729</math> and <math>b_{k+1}=b_4 = 1+3d = 271</math> and <math>c_{k+1}=c_4=a_4+b_4 = 729+271=1000</math>; so it works! Then <math>c_k = c_3 = 9^2+181 = 262</math>.
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== Solution 3==
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Using the same reasoning (<math>100</math> isn't very big), we can guess which terms will work. The first case is <math>k=3</math>, so we assume the second and fourth terms of <math>c</math> are <math>100</math> and <math>1000</math>. We let <math>r</math> be the common ratio of the geometric sequence and write the arithmetic relationships in terms of <math>r</math>.
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The common difference is <math>100-r - 1</math>, and so we can equate: <math>2(99-r)+100-r=1000-r^3</math>. Moving all the terms to one side and the constants to the other yields
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<math>r^3-3r = 702</math>, or <math>r(r^2-3) = 702</math>. Simply listing out the factors of <math>702</math> shows that the only factor <math>3</math> less than a square that works is <math>78</math>. Thus <math>r=9</math> and we solve from there to get <math>\boxed{262}</math>.
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Solution by rocketscience
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== Solution 4 (More Robust Bash) ==
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The reason for bashing in this context can also be justified by the fact 100 isn't very big.
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Let the common difference for the arithmetic sequence be <math>a</math>, and the common ratio for the geometric sequence be <math>b</math>. The sequences are now <math>1, a+1, 2a+1, \ldots</math>, and <math>1, b, b^2, \ldots</math>. We can now write the given two equations as the following:
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<math>1+(k-2)a+b^{k-2} = 100</math>
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<math>1+ka+b^k = 1000</math>
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Take the difference between the two equations to get <math>2a+(b^2-1)b^{k-2} = 900</math>. Since 900 is divisible by 4, we can tell <math>a</math> is even and <math>b</math> is odd. Let <math>a=2m</math>, <math>b=2n+1</math>, where <math>m</math> and <math>n</math> are positive integers. Substitute variables and divide by 4 to get:
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<math>m+(n+1)(n)(2n+1)^{k-2} = 225</math>
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Because very small integers for <math>n</math> yield very big results, we can bash through all cases of <math>n</math>. Here, we set an upper bound for <math>n</math> by setting <math>k</math> as 3. After trying values, we find that <math>n\leq 4</math>, so <math>b=9, 7, 5, 3</math>. Testing out <math>b=9</math> yields the correct answer of <math>\boxed{262}</math>. Note that even if this answer were associated with another b value like <math>b=3</math>, the value of <math>k</math> can still only be 3 for all of the cases.
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==Solution 5 (Casework)==
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Let <math>a_n</math> and <math>b_n </math> be in the form of
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<cmath>\begin{array}[b]{ c c c c c c c }
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n & 1 & 2 & 3 & 4 & 5 & 6 \\[2ex]
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a_n & 1 & a+1 & 2a+1 & 3a+1 & 4a+1 & 5a+1 \\[2ex]
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b_n & 1 & b & b^2 & b^3 & b^4 & b^5 \\[2ex]
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c_n & 2 & b+a+1 & b^2+2a+1 & b^3+3a+1 & b^4+4a+1 & b^5+5a+1
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\end{array} </cmath>
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Case <math>1.\hspace{10mm}  k = 3\hspace{5mm} (c_1=2 \implies k > 2).</math>
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<cmath>c_2 = a+1 + b = 100, c_4 = 3a+1 + b^3 = 1000 \implies b^3 -3b -2 = 1000-300 \implies  b^3 - 3b = 702 \implies b = 9, a=90, c_3 = \boxed {262}.</cmath>
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Case <math>2. \hspace{10mm}  k = 4.</math>
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<cmath>c_3 = 2a+1 + b^2 = 100, c_5 = 4a+1 + b^4 = 1000 \implies b^4 -2b^2 -1 = 1000-200 \implies  b^4 - 2b^2 = 801 \implies \O.</cmath>
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Case <math>3. \hspace{10mm}  k \ge 5.\hspace{3mm}  b^5 < 1000 \implies b = \{2,3\}.</math>
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Case <math>3.1.\hspace{10mm} b = 2.</math>
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<cmath>c_{k-1} = 2^{k-2} + (k-2) a + 1 = 100, c_{k+1} = 2^k + ka + 1 = 1000\implies a = 450-3\cdot 2^{k-3} \implies 2^k +450k -3k\cdot 2^{k-3} + 1 = 1000  \implies \O.</cmath>
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Case <math>3.2.\hspace{10mm} b = 3.</math>
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<cmath>c_{k-1} = 3^{k-2} + (k-2) a + 1 = 100, c_{k+1} = 3^k + ka + 1 = 1000\implies a = 450-4\cdot 3^{k-2} \implies 3^{k-4}(9-4k) +50k = 3\cdot 37  \implies \O.</cmath>
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'''vladimir.shelomovskii@gmail.com, vvsss'''
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== See also ==
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{{AIME box|year=2016|n=II|num-b=8|num-a=10}}
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{{MAA Notice}}

Latest revision as of 00:18, 2 February 2023

Problem

The sequences of positive integers $1,a_2, a_3,...$ and $1,b_2, b_3,...$ are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let $c_n=a_n+b_n$. There is an integer $k$ such that $c_{k-1}=100$ and $c_{k+1}=1000$. Find $c_k$.

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 $b_2$. When we get to $b_2=9$ and $a_2=91$, we have $a_4=271$ and $b_4=729$, which works, therefore, the answer is $b_3+a_3=81+181=\boxed{262}$.

Solution 2 (Some trial and error)

We have $a_k=r^{k-1}$ and $b_k=(k-1)d$. First, $b_{k-1}<c_{k-1}=100$ implies $d<100$, so $b_{k+1}<300$.

It follows that $a_{k+1}=1000-b_{k+1}>700$, i.e., \[700 < r^k < 1000.\] Moreover, since $k$ is atleast $3$ we get $r^3\le r^k <1000$, i.e. $r<10$. For every value of $r$ in this range, define $i(r) = \max \{x : r^x < 700\}$, and define $j(r) = \min \{x : r^x > 1000\}$. We are looking for values of $r$ such that $j(r) -i(r)>1$. Let's make a table: \begin{array}[b]{ c c c c c c c c c } r & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\[2ex] i(r) & 9 & 5 & 4 & 4 & 3 & 3 & 3 & 2\\[2ex] j(r) & 10 & 7 & 5 & 5 & 4 & 4 & 4 & 4 \end{array} The only admissible values for $r^k$ are $\{3^6, 9^3\}$. However, since $100=c_{k-1}=r^{k-2}+(k-2)d+1$, we must have $(k-2)\mid 99-r^{k-2}$. This does not hold for $r^k=3^6$ because $4$ does not divide $99-3^4=18$. This leaves $r^k=9^3$ as the only option.

For $r=9$ and $k=3$, we check: $a_{k-1}= a_2 = r =9$ implies $b_{k-1}= b_2 = 91$, i.e. $d=90$. Then $a_{k+1}=a_4 = r^3 = 729$ and $b_{k+1}=b_4 = 1+3d = 271$ and $c_{k+1}=c_4=a_4+b_4 = 729+271=1000$; so it works! Then $c_k = c_3 = 9^2+181 = 262$.

Solution 3

Using the same reasoning ($100$ isn't very big), we can guess which terms will work. The first case is $k=3$, so we assume the second and fourth terms of $c$ are $100$ and $1000$. We let $r$ be the common ratio of the geometric sequence and write the arithmetic relationships in terms of $r$.

The common difference is $100-r - 1$, and so we can equate: $2(99-r)+100-r=1000-r^3$. Moving all the terms to one side and the constants to the other yields

$r^3-3r = 702$, or $r(r^2-3) = 702$. Simply listing out the factors of $702$ shows that the only factor $3$ less than a square that works is $78$. Thus $r=9$ and we solve from there to get $\boxed{262}$.

Solution by rocketscience

Solution 4 (More Robust Bash)

The reason for bashing in this context can also be justified by the fact 100 isn't very big.

Let the common difference for the arithmetic sequence be $a$, and the common ratio for the geometric sequence be $b$. The sequences are now $1, a+1, 2a+1, \ldots$, and $1, b, b^2, \ldots$. We can now write the given two equations as the following:

$1+(k-2)a+b^{k-2} = 100$

$1+ka+b^k = 1000$

Take the difference between the two equations to get $2a+(b^2-1)b^{k-2} = 900$. Since 900 is divisible by 4, we can tell $a$ is even and $b$ is odd. Let $a=2m$, $b=2n+1$, where $m$ and $n$ are positive integers. Substitute variables and divide by 4 to get:

$m+(n+1)(n)(2n+1)^{k-2} = 225$

Because very small integers for $n$ yield very big results, we can bash through all cases of $n$. Here, we set an upper bound for $n$ by setting $k$ as 3. After trying values, we find that $n\leq 4$, so $b=9, 7, 5, 3$. Testing out $b=9$ yields the correct answer of $\boxed{262}$. Note that even if this answer were associated with another b value like $b=3$, the value of $k$ can still only be 3 for all of the cases.

Solution 5 (Casework)

Let $a_n$ and $b_n$ be in the form of \begin{array}[b]{ c c c c c c c } n & 1 & 2 & 3 & 4 & 5 & 6 \\[2ex] a_n & 1 & a+1 & 2a+1 & 3a+1 & 4a+1 & 5a+1 \\[2ex] b_n & 1 & b & b^2 & b^3 & b^4 & b^5 \\[2ex] c_n & 2 & b+a+1 & b^2+2a+1 & b^3+3a+1 & b^4+4a+1 & b^5+5a+1 \end{array} Case $1.\hspace{10mm} k = 3\hspace{5mm} (c_1=2 \implies k > 2).$ \[c_2 = a+1 + b = 100, c_4 = 3a+1 + b^3 = 1000 \implies b^3 -3b -2 = 1000-300 \implies b^3 - 3b = 702 \implies b = 9, a=90, c_3 = \boxed {262}.\] Case $2. \hspace{10mm} k = 4.$ \[c_3 = 2a+1 + b^2 = 100, c_5 = 4a+1 + b^4 = 1000 \implies b^4 -2b^2 -1 = 1000-200 \implies b^4 - 2b^2 = 801 \implies \O.\] Case $3. \hspace{10mm} k \ge 5.\hspace{3mm} b^5 < 1000 \implies b = \{2,3\}.$

Case $3.1.\hspace{10mm} b = 2.$ \[c_{k-1} = 2^{k-2} + (k-2) a + 1 = 100, c_{k+1} = 2^k + ka + 1 = 1000\implies a = 450-3\cdot 2^{k-3} \implies 2^k +450k -3k\cdot 2^{k-3} + 1 = 1000 \implies \O.\] Case $3.2.\hspace{10mm} b = 3.$ \[c_{k-1} = 3^{k-2} + (k-2) a + 1 = 100, c_{k+1} = 3^k + ka + 1 = 1000\implies a = 450-4\cdot 3^{k-2} \implies 3^{k-4}(9-4k) +50k = 3\cdot 37 \implies \O.\] vladimir.shelomovskii@gmail.com, vvsss

See also

2016 AIME II (Problems • Answer Key • Resources)
Preceded by
Problem 8 		Followed by
Problem 10
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