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

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Therefore, the minimum sum of our infinite geometric sequence is <math>\boxed{\textbf{(E)}\ 4}</math>.
 
Therefore, the minimum sum of our infinite geometric sequence is <math>\boxed{\textbf{(E)}\ 4}</math>.
  
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==Solution 2==
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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>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>1/x</math>.
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With further observation we realize that in order for the 2nd term to be <math>1</math>, the first term has to be <math>x</math>. So than in order to minimize the sum, we minimize have to <math>x</math>.
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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>1/2</math>. Thus the sum is <math>\frac{2}{1/2}</math> or <math>\boxed{\textbf{(E)}\ 4}</math>.
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Solution 2 by I_Dont_Do_Math
 
(Solution edited by akaashp11)
 
(Solution edited by akaashp11)
  

Revision as of 13:56, 21 February 2016

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

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: \[\begin{split} a_1*r & = 1 \\ a_1 & = \frac{1}{r} \end{split}\]

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 negative $a$ 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: \[\begin{split} S & = \frac{1}{\frac{1}{2} -(\frac{1}{2})^2} \\ S & = \frac{1}{\frac{1}{4}}  \end{split}\]

Therefore, the minimum sum of our infinite geometric sequence 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 $1/x$ 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 $1/x$.

With further observation we realize that in order for the 2nd term to be $1$, the first term has to be $x$. So than in order to minimize the sum, we minimize have to $x$.

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 $1/2$. Thus the sum is $\frac{2}{1/2}$ or $\boxed{\textbf{(E)}\ 4}$. Solution 2 by I_Dont_Do_Math (Solution edited by akaashp11)

See Also

2016 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 15
Followed by
Problem 17
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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