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Difference between revisions of "2016 AMC 10B Problems/Problem 16"

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\textbf{(E)}\ 4</math>
 
\textbf{(E)}\ 4</math>
  
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==Solution 1==
−
==Solution==
 
 
The sum of an infinite geometric series is of the form:
 
The sum of an infinite geometric series is of the form:
−
S=\frac{a_1}{1-r}.
+
<cmath>\begin{split}
 +
S & = \frac{a_1}{1-r}  
 +
\end{split}</cmath>
 
where <math>a_1</math> is the first term and <math>r</math> is the ratio whose absolute value is less than 1.
 
where <math>a_1</math> is the first term and <math>r</math> is the ratio whose absolute value is less than 1.
 +
 
We know that the second term is the first term multiplied by the ratio.  
 
We know that the second term is the first term multiplied by the ratio.  
 
In other words:
 
In other words:
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<math>a_1 \cdot r=1</math>
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<cmath>\begin{split}
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<math>a_1=\frac{1}{r}</math>
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a_1 \cdot r & = 1 \\
−
Thus the sum is the following:
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a_1 & = \frac{1}{r}
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<math>S=\frac{\frac{1}{r}}{1-r}</math>
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\end{split}</cmath>
−
We can multiply <math>r</math> to both sides of the numerator and denominator.
+
 
−
<math>S=\frac{1}{r-r^2}</math>
+
Thus, the sum is the following:
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Since we want the minimum value of this expression, we want the maximum value for the denominator.
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<cmath>\begin{split}
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<math>max(-r^2+r)</math>
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S & = \frac{\frac{1}{r}}{1-r} \\\\
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The maximum value of a quadratic with negative <math>a</math> is <math>\frac{-b}{2a}</math>.
+
S & =\frac{1}{r-r^2}
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<math>S=\frac{-(1)}{2(-1)}=\frac{1}{2}</math>
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\end{split}</cmath>
−
Plugging 1/2 in, we get:
+
 
−
<math>S=\frac{1}{\frac{1}{2}}=2</math>, <math>B</math>.
+
Since we want the minimum value of this expression, we want the maximum value for the denominator, <math>-r^2+r</math>.
 +
The maximum x-value of a quadratic with leading coefficient <math>-a</math> is <math>\frac{-b}{2a}</math>.
 +
<cmath>\begin{split}
 +
r & = \frac{-(1)}{2(-1)} \\\\
 +
r & = \frac{1}{2}
 +
\end{split}</cmath>
 +
 
 +
Plugging <math>r</math> <math>=</math> <math>\frac{1}{2}</math> into the quadratic yields:
 +
<cmath>\begin{split}
 +
S & = \frac{1}{\frac{1}{2} -\left(\frac{1}{2}\right)^2} \\\\
 +
S & = \frac{1}{\frac{1}{4}}
 +
\end{split}</cmath>
 +
 
 +
Therefore, the minimum sum of our infinite geometric sequence is <math>\boxed{\textbf{(E)}\ 4}</math>.
 +
(Solution by akaashp11)
 +
 
 +
As an extension to find the maximum value for the denominator we can find the derivative of <math>-r^2+r</math> to get <math>1-2r</math>. we know that this changes sign when <math>r = \frac{1}{2}</math> so plugging it in into the original equation we find the answer is <math>\boxed{\textbf{(E)}\ 4}</math>.
 +
 
 +
==Solution 2==
 +
After observation we realize that in order to minimize our sum <math>\frac{a}{1-r}</math> with <math>a</math> being the reciprocal of r, the common ratio <math>r</math> has to be in the form of <math>\frac{1}{x}</math>, with <math>x</math> being an integer, as anything more than <math>1</math> divided by <math>x</math> would give a larger sum than a ratio in the form of <math>\frac{1}{x}</math>.
 +
 
 +
The first term has to be <math>x</math>, so then in order to minimize the sum, we have minimize <math>x</math>.
 +
 
 +
The smallest possible value for <math>x</math> such that it is an integer that's greater than <math>1</math> is <math>2</math>. So our first term is <math>2</math> and our common ratio is <math>\frac{1}{2}</math>. Thus the sum is <math>\frac{2}{\frac {1}{2}}</math> or <math>\boxed{\textbf{(E)}\ 4}</math>.
 +
Solution 2 by No_One
 +
 
 +
(edited)
 +
 
 +
==Solution 3==
 +
We can see that if <math>a</math> is the first term, and <math>r</math> is the common ratio between each of the terms, then we can get
 +
<cmath>S=\frac{a}{1-r} \implies S-Sr=a</cmath>
 +
Also, we know that the second term can be expressed as <math>a\cdot r</math>
 +
notice if we multiply <math>S-Sr=a</math> by <math>r</math>, we would get
 +
<cmath>r(S-Sr)=ar \implies Sr-Sr^2=1 \implies Sr^2-Sr+1=0</cmath>
 +
This quadratic has real solutions if the discriminant is greater than or equal to zero, or
 +
<cmath>S^2-4\cdot S \cdot 1 \ge 0</cmath>
 +
This yields that <math>S\le 0</math> or <math>S\ge 4</math>.
 +
However, since we know that <math>S</math> has to be positive, we can safely conclude that the minimum possible value of <math>S</math> is <math>\boxed{\textbf{(E)}\ 4}</math>.
 +
 
 +
==Solution 4 (Quick Method)==
 +
Let the first term of the geometric series <math>x</math>. Since it must be decreasing, we have <math>x>1</math> and the third term is <math>
 +
\frac{1}{x}</math>. Realize that by AM-GM inequality <math>x+\frac{1}{x} \ge 2</math> with equality if <math>x = 1</math>. However, we established that <math>x>1</math> so that means <math>x+\frac{1}{x} > 2</math>. So the sum of the first three terms of the sequence <math>x + \frac{1}{x} + 1</math> is greater than <math>3</math>, and the geometric series keeps continuing  infinitely.  This means the sum continues increasing. The only answer choice greater than <math>3</math> is <math>\boxed{\textbf{(E)}\ 4}</math>. ~skyscraper
 +
 
 +
 
 +
== Solution 5 (Clever Algebra) ==
 +
 
 +
Let the first term be <math>k.</math> The sum of the series is <math>\frac{k}{1- \frac{1}{k}} =\frac{k^2}{k-1}.</math> Rewrite this as <math>\frac{k^2-2k+1}{k-1} +\frac{2k-1}{k-1} = k-1+\frac{2k-2}{k-1} +\frac{1}{k-1} = (k-1) + \left(\frac{1}{k-1}\right) + 2.</math> By AM-GM we know that <math>(k-1) + \left(\frac{1}{k-1}\right) \ge 2</math> so the minimum is <math>2+2 = \boxed{\textbf{(E)}\ 4}.</math>
 +
 
 +
 
 +
== Solution 6 (Calculus) ==
 +
Set the first term is <math>a.</math>, the common ratio should be <math>\frac{1}{a}.</math>
 +
 
 +
The sum to infinity of the series is <math>S=\frac{a}{1-\frac{1}{a}}=\frac{a^2}{a-1}.</math>
 +
 
 +
Since <math>S</math> is positive, we have <math>a>1.</math>  Define the function <math>f(a)=\frac{a^2}{a-1}</math> , the domain of this function is <math>a>1.</math>
 +
 
 +
Let <math>f^{'}(a)=\frac{2a^2-2a-a^2}{(a-1)^2}=\frac{a(a-2)}{(a-1)^2}=0.</math> We solve that <math>a=2.</math>
 +
 
 +
It's easy to find that when <math>1<a<2, f^{'}(a)<0,</math> when <math>a>2, f^{'}(a)>0.</math> Thus <math>f(a)</math> has a minimum value when <math>a=2.</math>, which is <math>f(2)=4.</math> Choose  <math>\boxed{\textbf{(E)}\ 4}.</math>
 +
 
 +
~PythZhou
  
 
==See Also==
 
==See Also==
 
{{AMC10 box|year=2016|ab=B|num-b=15|num-a=17}}
 
{{AMC10 box|year=2016|ab=B|num-b=15|num-a=17}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 21:13, 2 November 2024

Problem

The sum of an infinite geometric series is a positive number $S$, and the second term in the series is $1$. What is the smallest possible value of $S?$

$\textbf{(A)}\ \frac{1+\sqrt{5}}{2} \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ \sqrt{5} \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$

Solution 1

The sum of an infinite geometric series is of the form: \[\begin{split} S & = \frac{a_1}{1-r} \end{split}\] where $a_1$ is the first term and $r$ is the ratio whose absolute value is less than 1.

We know that the second term is the first term multiplied by the ratio. In other words: \[\begin{split} a_1 \cdot r & = 1 \\ a_1 & = \frac{1}{r} \end{split}\]

Thus, the sum is the following: \[\begin{split} S & = \frac{\frac{1}{r}}{1-r} \\\\ S & =\frac{1}{r-r^2} \end{split}\]

Since we want the minimum value of this expression, we want the maximum value for the denominator, $-r^2+r$. The maximum x-value of a quadratic with leading coefficient $-a$ is $\frac{-b}{2a}$. \[\begin{split} r & = \frac{-(1)}{2(-1)} \\\\ r & = \frac{1}{2} \end{split}\]

Plugging $r$ $=$ $\frac{1}{2}$ into the quadratic yields: \[\begin{split} S & = \frac{1}{\frac{1}{2} -\left(\frac{1}{2}\right)^2} \\\\ S & = \frac{1}{\frac{1}{4}} \end{split}\]

Therefore, the minimum sum of our infinite geometric sequence is $\boxed{\textbf{(E)}\ 4}$. (Solution by akaashp11)

As an extension to find the maximum value for the denominator we can find the derivative of $-r^2+r$ to get $1-2r$. we know that this changes sign when $r = \frac{1}{2}$ so plugging it in into the original equation we find the answer is $\boxed{\textbf{(E)}\ 4}$.

Solution 2

After observation we realize that in order to minimize our sum $\frac{a}{1-r}$ with $a$ being the reciprocal of r, the common ratio $r$ has to be in the form of $\frac{1}{x}$, with $x$ being an integer, as anything more than $1$ divided by $x$ would give a larger sum than a ratio in the form of $\frac{1}{x}$.

The first term has to be $x$, so then in order to minimize the sum, we have minimize $x$.

The smallest possible value for $x$ such that it is an integer that's greater than $1$ is $2$. So our first term is $2$ and our common ratio is $\frac{1}{2}$. Thus the sum is $\frac{2}{\frac {1}{2}}$ or $\boxed{\textbf{(E)}\ 4}$. Solution 2 by No_One

(edited)

Solution 3

We can see that if $a$ is the first term, and $r$ is the common ratio between each of the terms, then we can get \[S=\frac{a}{1-r} \implies S-Sr=a\] Also, we know that the second term can be expressed as $a\cdot r$ notice if we multiply $S-Sr=a$ by $r$, we would get \[r(S-Sr)=ar \implies Sr-Sr^2=1 \implies Sr^2-Sr+1=0\] This quadratic has real solutions if the discriminant is greater than or equal to zero, or \[S^2-4\cdot S \cdot 1 \ge 0\] This yields that $S\le 0$ or $S\ge 4$. However, since we know that $S$ has to be positive, we can safely conclude that the minimum possible value of $S$ is $\boxed{\textbf{(E)}\ 4}$.

Solution 4 (Quick Method)

Let the first term of the geometric series $x$. Since it must be decreasing, we have $x>1$ and the third term is $\frac{1}{x}$. Realize that by AM-GM inequality $x+\frac{1}{x} \ge 2$ with equality if $x = 1$. However, we established that $x>1$ so that means $x+\frac{1}{x} > 2$. So the sum of the first three terms of the sequence $x + \frac{1}{x} + 1$ is greater than $3$, and the geometric series keeps continuing infinitely. This means the sum continues increasing. The only answer choice greater than $3$ is $\boxed{\textbf{(E)}\ 4}$. ~skyscraper


Solution 5 (Clever Algebra)

Let the first term be $k.$ The sum of the series is $\frac{k}{1- \frac{1}{k}} =\frac{k^2}{k-1}.$ Rewrite this as $\frac{k^2-2k+1}{k-1} +\frac{2k-1}{k-1} = k-1+\frac{2k-2}{k-1} +\frac{1}{k-1} = (k-1) + \left(\frac{1}{k-1}\right) + 2.$ By AM-GM we know that $(k-1) + \left(\frac{1}{k-1}\right) \ge 2$ so the minimum is $2+2 = \boxed{\textbf{(E)}\ 4}.$


Solution 6 (Calculus)

Set the first term is $a.$, the common ratio should be $\frac{1}{a}.$

The sum to infinity of the series is $S=\frac{a}{1-\frac{1}{a}}=\frac{a^2}{a-1}.$

Since $S$ is positive, we have $a>1.$ Define the function $f(a)=\frac{a^2}{a-1}$ , the domain of this function is $a>1.$

Let $f^{'}(a)=\frac{2a^2-2a-a^2}{(a-1)^2}=\frac{a(a-2)}{(a-1)^2}=0.$ We solve that $a=2.$

It's easy to find that when $1<a<2, f^{'}(a)<0,$ when $a>2, f^{'}(a)>0.$ Thus $f(a)$ has a minimum value when $a=2.$, which is $f(2)=4.$ Choose $\boxed{\textbf{(E)}\ 4}.$

~PythZhou

See Also

2016 AMC 10B (Problems • Answer Key • Resources)
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Problem 15 		Followed by
Problem 17
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