Difference between revisions of "1969 AHSME Problems/Problem 16"
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== Solution == | == Solution == | ||
| − | <math>\fbox{E}</math> | + | Since <math>a=kb</math>, we can write <math>(a-b)^n</math> as <math>(kb-b)^n</math>. |
| + | Expanding, the second term is <math>-k^{n-1}b^{n}{{n}\choose{1}}</math>, and the third term is <math>k^{n-2}b^{n}{{n}\choose{2}}</math>, so we can write the equation | ||
| + | <cmath>-k^{n-1}b^{n}{{n}\choose{1}}+k^{n-2}b^{n}{{n}\choose{2}}=0</cmath> | ||
| + | Simplifying and multiplying by two to remove the denominator, we get | ||
| + | <cmath>-2k^{n-1}b^{n}n+k^{n-2}b^{n}n(n-1)=0</cmath> | ||
| + | Factoring, we get | ||
| + | <cmath>k^{n-2}b^{n}n(-2k+n-1)=0</cmath> | ||
| + | Dividing by <math>k^{n-2}b^{n}</math> gives | ||
| + | <cmath>n(-2k+n-1)=0</cmath> | ||
| + | Since it is given that <math>n\ge2</math>, <math>n</math> cannot equal 0, so we can divide by n, which gives | ||
| + | <cmath>-2k+n-1=0</cmath> | ||
| + | Solving for <math>n</math> gives | ||
| + | <cmath>n=2k+1</cmath> so the answer is <math>\fbox{E}</math>. | ||
== See also == | == See also == | ||
Latest revision as of 12:16, 11 October 2015
Problem
When 
, is expanded by the binomial theorem, it is found that when 
, where 
is a positive integer, the sum of the second and third terms is zero. Then 
equals:
Solution
Since , we can write 
as 
.
Expanding, the second term is 
, and the third term is 
, so we can write the equation
![\[-k^{n-1}b^{n}{{n}\choose{1}}+k^{n-2}b^{n}{{n}\choose{2}}=0\]](http://latex.artofproblemsolving.com/b/4/c/b4ce32d058aa729992586d21038a517ae9798036.png)
Simplifying and multiplying by two to remove the denominator, we get
![\[-2k^{n-1}b^{n}n+k^{n-2}b^{n}n(n-1)=0\]](http://latex.artofproblemsolving.com/f/9/3/f939c798b9d592410347014b8f3437ced4432412.png)
Factoring, we get
![\[k^{n-2}b^{n}n(-2k+n-1)=0\]](http://latex.artofproblemsolving.com/1/6/a/16a45a2f061795398abf0c75ba84389dc00bf1f3.png)
Dividing by 
gives
![\[n(-2k+n-1)=0\]](http://latex.artofproblemsolving.com/1/b/2/1b2da0bf65461a55a5dd31e9036e1d68b538ee6c.png)
Since it is given that 
, cannot equal 0, so we can divide by n, which gives
![\[-2k+n-1=0\]](http://latex.artofproblemsolving.com/9/e/2/9e27d9d86c519ef42e19a28281463d330165d6eb.png)
Solving for gives
![\[n=2k+1\]](http://latex.artofproblemsolving.com/5/f/9/5f9fafce7623c22a41a4db837ea5cefeb431bd6a.png)
so the answer is 
.
See also
| 1969 AHSC (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 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
