Difference between revisions of "1969 AHSME Problems/Problem 32"
(Created page with "== Problem == Let a sequence <math>\{u_n\}</math> be defined by <math>u_1=5</math> and the relationship <math>u_{n+1}-u_n=3+4(n-1), n=1,2,3\cdots.</math>If <math>u_n</math> is e...") |
Rockmanex3 (talk | contribs) (Solution to Problem 32) |
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== Solution == | == Solution == | ||
− | <math>\ | + | |
+ | 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>12</math>, and <math>23</math>. From there, a [[system of equations]] can be written. | ||
+ | <cmath>a+b+c=5</cmath> | ||
+ | <cmath>4a+2b+c=12</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>. | ||
== See also == | == See also == |
Revision as of 04:06, 10 June 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.
Solve the system to get
,
, and
. The sum of the coefficients is
.
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.