Difference between revisions of "1971 AHSME Problems/Problem 29"
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− | The product of the sequence <math>10^{\dfrac{1}{11}}, 10^{\dfrac{2}{11}}, 10^{\dfrac{3}{11}}, 10^{\dfrac{4}{11}},\dots , 10^{\dfrac{n}{11}}</math> is equal to <math>10^{\dfrac{1}{11}+\frac{2}{11}\dots\frac{n}{11}}</math> since we are looking for the smallest value <math>n</math> that will create <math>100,000</math>, or <math>10^5</math>, we set up the equation <math>10^{\dfrac{1}{11}+\frac{2}{11}\dots\frac{n}{11}}=10^5</math> | + | The product of the sequence <math>10^{\dfrac{1}{11}}, 10^{\dfrac{2}{11}}, 10^{\dfrac{3}{11}}, 10^{\dfrac{4}{11}},\dots , 10^{\dfrac{n}{11}}</math> is equal to <math>10^{\dfrac{1}{11}+\frac{2}{11}\dots\frac{n}{11}}</math> since we are looking for the smallest value <math>n</math> that will create <math>100,000</math>, or <math>10^5</math>, we set up the equation <math>10^{\dfrac{1}{11}+\frac{2}{11}\dots\frac{n}{11}}=10^5</math>, which simplified to <math>\dfrac{1}{11}+\frac{2}{11}\dots\frac{n}{11}=5</math>, or <math>1+2+3\dots n=55</math> This can be converted to <math>\frac{n(1+n)}{2}=5</math> This simplified to the quadratic <math>n^2+n-10=0</math> |
Revision as of 17:16, 22 August 2019
Problem 29
Given the progression . The least positive integer such that the product of the first terms of the progression exceeds is
Solution
The product of the sequence is equal to since we are looking for the smallest value that will create , or , we set up the equation , which simplified to , or This can be converted to This simplified to the quadratic