Difference between revisions of "1992 AIME Problems/Problem 8"
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== Problem == | == Problem == | ||
− | For any sequence of real numbers <math>A=(a_1,a_2,a_3,\ldots)</math>, define <math>\Delta A^{}_{}</math> to be the sequence <math>(a_2-a_1,a_3-a_2,a_4-a_3,\ldots)</math>, whose <math>n^\mbox{th}_{}</math> term is <math>a_{n+1}-a_n^{}</math>. Suppose that all of the terms of the sequence <math>\Delta(\Delta A^{}_{})</math> are <math>1^{}_{}</math>, and that <math>a_{19}=a_{92}^{}=0</math>. Find <math>a_1^{}</math>. | + | For any sequence of real numbers <math>A=(a_1,a_2,a_3,\ldots)</math>, define <math>\Delta A^{}_{}</math> to be the sequence <math>(a_2-a_1,a_3-a_2,a_4-a_3,\ldots)</math>, whose <math>n^{\mbox{th}}_{}</math> term is <math>a_{n+1}-a_n^{}</math>. Suppose that all of the terms of the sequence <math>\Delta(\Delta A^{}_{})</math> are <math>1^{}_{}</math>, and that <math>a_{19}=a_{92}^{}=0</math>. Find <math>a_1^{}</math>. |
== Solution == | == Solution == | ||
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== Solution 3 == | == Solution 3 == | ||
− | + | The sequence <math>\Delta(\Delta A)</math> is the second finite difference sequence, and the first <math>k-1</math> terms of this sequence can be computed in terms of the original sequence as shown below. | |
− | <math>a_3+a_1-2 | + | <math>\begin{array}{rcl} |
+ | a_3+a_1-2a_2&=&1\\ | ||
+ | a_4+a_2-2a_3&=&1\\ | ||
+ | &\vdots\\ | ||
+ | a_k + a_{k-2} - 2a_{k-1} &= &1\\ | ||
+ | a_{k+1} + a_{k-1} - 2a_k &=& 1.\\ | ||
+ | \end{array} | ||
+ | </math> | ||
− | <math> | + | Adding the above <math>k-1</math> equations we find that |
− | + | <cmath>(a_{k+1} - a_k) = k-1 + (a_2-a_1).\tag{1}</cmath> | |
− | + | We can sum equation <math>(1)</math> from <math>k=1</math> to <math>18</math>, finding | |
+ | <cmath>18(a_1-a_2) - a_1 = 153.\tag{2} </cmath> | ||
− | + | We can also sum equation <math>(1)</math> from <math>k=1</math> to <math>91</math>, finding | |
− | + | <cmath>91(a_1-a_2) - a_1 = 4095.\tag{3}</cmath> | |
− | + | Finally, <math>18\cdot (3) - 91\cdot(2)</math> gives <math>a_1=\boxed{819}</math>. | |
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− | <math>18 | ||
Kris17 | Kris17 |
Revision as of 00:39, 7 March 2015
Problem
For any sequence of real numbers , define to be the sequence , whose term is . Suppose that all of the terms of the sequence are , and that . Find .
Solution
Note that the s are reminiscent of differentiation; from the condition , we are led to consider the differential equation This inspires us to guess a quadratic with leading coefficient 1/2 as the solution; as we must have roots at and .
Thus, .
Solution 2
Let , and .
Note that in every sequence of ,
Then
Since ,
Solving, .
Solution 3
The sequence is the second finite difference sequence, and the first terms of this sequence can be computed in terms of the original sequence as shown below.
Adding the above equations we find that
We can sum equation from to , finding
We can also sum equation from to , finding Finally, gives .
Kris17
See also
1992 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.