Difference between revisions of "1969 AHSME Problems/Problem 32"
Rockmanex3 (talk | contribs) (Solution to Problem 32) |
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<cmath>9a+3b+c=23</cmath> | <cmath>9a+3b+c=23</cmath> | ||
Solve the system to get <math>a=2</math>, <math>b=1</math>, and <math>c=2</math>. The sum of the coefficients is <math>\boxed{\textbf{(C) } 5}</math>. | Solve the system to get <math>a=2</math>, <math>b=1</math>, and <math>c=2</math>. The sum of the coefficients is <math>\boxed{\textbf{(C) } 5}</math>. | ||
| + | |||
| + | Note: solving the system is extra work, as the answer is described by the first equation. | ||
== See also == | == See also == | ||
Revision as of 13:24, 21 November 2018
Problem
Let a sequence 
be defined by 
and the relationship 
If 
is expressed as a polynomial in 
, the algebraic sum of its coefficients is:
Solution
Note that the first differences create a linear function, so the sequence 
is quadratic.
The first three terms of the sequence are 
, 
, and 
. From there, a system of equations can be written.
![\[a+b+c=5\]](http://latex.artofproblemsolving.com/8/8/5/885ecad0cde51b771e9628de89581ec61ddc57d4.png)
![\[4a+2b+c=12\]](http://latex.artofproblemsolving.com/7/a/b/7abc4453750cbecffc1baf0a1cf797bdd8d9db3a.png)
![\[9a+3b+c=23\]](http://latex.artofproblemsolving.com/6/f/f/6ff50f7e9066104271369883281b29fd04a25f66.png)
Solve the system to get 
, 
, and 
. The sum of the coefficients is 
.
Note: solving the system is extra work, as the answer is described by the first equation.
See also
| 1969 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 31 |
Followed by Problem 33 | |
| 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. 
