Difference between revisions of "1969 AHSME Problems/Problem 32"
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Note that the first differences create a linear function, so the sequence <math>{u_n}</math> is quadratic. | Note that the first differences create a linear function, so the sequence <math>{u_n}</math> is quadratic. | ||
| − | The first three terms of the sequence are <math>5</math>, <math> | + | The first three terms of the sequence are <math>5</math>, <math>8</math>, and <math>15</math>. From there, a [[system of equations]] can be written. |
<cmath>a+b+c=5</cmath> | <cmath>a+b+c=5</cmath> | ||
| − | <cmath>4a+2b+c= | + | <cmath>4a+2b+c=8</cmath> |
| − | <cmath>9a+3b+c= | + | <cmath>9a+3b+c=15</cmath> |
| − | Solve the system to get <math>a=2</math>, <math>b= | + | Solve the system to get <math>a=2</math>, <math>b=-3</math>, and <math>c=6</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. The sum of the coefficients (<math>a + b + c</math>) is just 5 by the first equation. | ||
== See also == | == See also == | ||
Latest revision as of 03:19, 14 July 2019
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)
![\[4a+2b+c=8\]](http://latex.artofproblemsolving.com/2/4/4/2447698ebfc6b198950c165df06ba51e522c37e3.png)
![\[9a+3b+c=15\]](http://latex.artofproblemsolving.com/b/8/7/b87c5b190645bd5c57646f4f480b8f60fa4f0139.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. The sum of the coefficients () is just 5 by the first equation.
) is just 5 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. 
