Difference between revisions of "1989 AIME Problems/Problem 8"
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Find the value of <math>16x_1+25x_2+36x_3+49x_4+64x_5+81x_6+100x_7^{}</math>. | Find the value of <math>16x_1+25x_2+36x_3+49x_4+64x_5+81x_6+100x_7^{}</math>. | ||
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== Solution == | == Solution == | ||
=== Solution 1=== | === Solution 1=== | ||
Revision as of 13:59, 25 November 2007
Problem
Assume that 
are real numbers such that
Find the value of 
.
Solution
Solution 1
Let us try to derive a way to find the last expression in terms of the three given expressions. The coefficients of 
in the first equation can be denoted as 
, making its coefficients in the second equation as 
and the third as 
. We need to find a way to sum them up to make 
.
Thus, we can write that 
. FOILing out all of the terms, we get 
. We can set up the three equation system:
Subtracting the second and third equations yields that 
, so 
and 
. Thus, we have to add 
.
Solution 2
Notice that we may rewrite the equations in the more compact form:
and 
, where 
and 
and 
is what we're trying to find.
Now undergo a paradigm shift: consider the polynomial in 
(we are only treating the as coefficients).
Notice that the degree of 
must be 
; it is a quadratic. We are given 
as 
and are asked to find . Using the concept of finite differences (a prototype of differentiation) we find that the second differences of consecutive values is constant, so that by arithmetic operations we find 
.
See also
| 1989 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 | ||
