2016 AMC 10B Problems/Problem 16

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 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: $$\begin{split} S & = \frac{1}{\frac{1}{2} -\left(\frac{1}{2}\right)^2} \\ S & = \frac{1}{\frac{1}{4}} \end{split}$$

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 $1/x$ 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 $1/x$.

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

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 $\boxed{\textbf{(E)}\ 4}$.