Difference between revisions of "2016 AMC 10B Problems/Problem 16"
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\end{split}</cmath> | \end{split}</cmath> | ||
− | Since we want the minimum value of this expression, we want the maximum value for the denominator, <math>-r^2 | + | 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 | + | The maximum x-value of a quadratic with leading coefficient <math>-a</math> is <math>\frac{-b}{2a}</math>. |
<cmath>\begin{split} | <cmath>\begin{split} | ||
r & = \frac{-(1)}{2(-1)} \\\\ | r & = \frac{-(1)}{2(-1)} \\\\ | ||
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(Solution by akaashp11) | (Solution by akaashp11) | ||
− | As an extension to find the maximum value for the denominator we can find the derivative of <math>-r^2 | + | 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== | ==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>1 | + | 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 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>1 | + | 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 | Solution 2 by No_One | ||
+ | |||
+ | (edited) | ||
==Solution 3== | ==Solution 3== | ||
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==Solution 4 (Quick Method)== | ==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> | 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 | + | \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 14:59, 12 June 2022
Contents
Problem
The sum of an infinite geometric series is a positive number , and the second term in the series is
. What is the smallest possible value of
Solution 1
The sum of an infinite geometric series is of the form:
where
is the first term and
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:
Thus, the sum is the following:
Since we want the minimum value of this expression, we want the maximum value for the denominator, .
The maximum x-value of a quadratic with leading coefficient
is
.
Plugging
into the quadratic yields:
Therefore, the minimum sum of our infinite geometric sequence is .
(Solution by akaashp11)
As an extension to find the maximum value for the denominator we can find the derivative of to get
. we know that this changes sign when
so plugging it in into the original equation we find the answer is
.
Solution 2
After observation we realize that in order to minimize our sum with
being the reciprocal of r. The common ratio
has to be in the form of
with
being an integer as anything more than
divided by
would give a larger sum than a ratio in the form of
.
The first term has to be , so then in order to minimize the sum, we have minimize
.
The smallest possible value for such that it is an integer that's greater than
is
. So our first term is
and our common ratio is
. Thus the sum is
or
.
Solution 2 by No_One
(edited)
Solution 3
We can see that if is the first term, and
is the common ratio between each of the terms, then we can get
Also, we know that the second term can be expressed as
notice if we multiply
by
, we would get
This quadratic has real solutions if the discriminant is greater than or equal to zero, or
This yields that
or
.
However, since we know that
has to be positive, we can safely conclude that the minimum possible value of
is
.
Solution 4 (Quick Method)
Let the first term of the geometric series . Since it must be decreasing, we have
and the third term is
. Realize that by AM-GM inequality
with equality if
. However, we established that
so that means
. So the sum of the first three terms of the sequence
is greater than
, and the geometric series keeps continuing infinitely. This means the sum continues increasing. The only answer choice greater than
is
. ~skyscraper
Solution 5 (Clever Algebra)
Let the first term be The sum of the series is
Rewrite this as
By AM-GM we know that
so the minimum is
Solution 6 (Calculus)
Set the first term is , the common ratio should be
The sum to infinity of the series is
Since is positive, we have
Define the function
, the domain of this function is
Let We solve that
It's easy to find that when when
Thus
has a minimum value when
, which is
Choose
~PythZhou
See Also
2016 AMC 10B (Problems • Answer Key • Resources) | ||
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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