Difference between revisions of "2016 AIME I Problems/Problem 1"
Firebolt360 (talk | contribs) (→Solution 1) |
Firebolt360 (talk | contribs) (→Solution 2) |
||
Line 1: | Line 1: | ||
==Problem 1== | ==Problem 1== | ||
For <math>-1<r<1</math>, let <math>S(r)</math> denote the sum of the geometric series <cmath>12+12r+12r^2+12r^3+\cdots .</cmath> Let <math>a</math> between <math>-1</math> and <math>1</math> satisfy <math>S(a)S(-a)=2016</math>. Find <math>S(a)+S(-a)</math>. | For <math>-1<r<1</math>, let <math>S(r)</math> denote the sum of the geometric series <cmath>12+12r+12r^2+12r^3+\cdots .</cmath> Let <math>a</math> between <math>-1</math> and <math>1</math> satisfy <math>S(a)S(-a)=2016</math>. Find <math>S(a)+S(-a)</math>. | ||
− | ==Solution | + | ==Solution== |
The sum of an infinite geometric series is <math>\frac{a}{1-r}\rightarrow \frac{12}{1\mp a}</math>. The product <math>S(a)S(-a)=\frac{144}{1-a^2}=2016</math> so dividing by <math>144</math> gives <math>\frac{1}{1-a^2}=14\implies a= \pm \sqrt{\frac{13}{14}}</math>. <math>\frac{12}{1-a}+\frac{12}{1+a}=\frac{24}{1-a^2}</math>, so the answer is <math>14\cdot 24=\boxed{336}</math>. | The sum of an infinite geometric series is <math>\frac{a}{1-r}\rightarrow \frac{12}{1\mp a}</math>. The product <math>S(a)S(-a)=\frac{144}{1-a^2}=2016</math> so dividing by <math>144</math> gives <math>\frac{1}{1-a^2}=14\implies a= \pm \sqrt{\frac{13}{14}}</math>. <math>\frac{12}{1-a}+\frac{12}{1+a}=\frac{24}{1-a^2}</math>, so the answer is <math>14\cdot 24=\boxed{336}</math>. | ||
Revision as of 14:17, 17 February 2019
Problem 1
For , let denote the sum of the geometric series Let between and satisfy . Find .
Solution
The sum of an infinite geometric series is . The product so dividing by gives . , so the answer is .
See also
2016 AIME I (Problems • Answer Key • Resources) | ||
Preceded by First Problem |
Followed by Problem 2 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.