Difference between revisions of "1998 AIME Problems/Problem 8"

Latest revision as of 12:11, 25 November 2024

Problem

Except for the first two terms, each term of the sequence $1000, x, 1000 - x,\ldots$ is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encounted. What positive integer $x$ produces a sequence of maximum length?

Solution

The best way to start is to just write out some terms.

0	1	2	3	4	5	6
$\quad 1000 \quad$ aa	$\quad x \quad$ aaa	$1000 - x$	$2x - 1000$ a	$2000 - 3x$	$5x - 3000$	$5000 - 8x$

It is now apparent that each term can be written as

$n \equiv 0 \pmod{2}\quad\quad F_{n-1}\cdot 1000-F_n\cdot x$
$n \equiv 1 \pmod{2}\quad\quad F_{n}\cdot 1000-F_{n-1}\cdot x$

where the $F_{n}$ are Fibonacci numbers. This can be proven through induction.

Solution 1

We can start to write out some of the inequalities now:

$x > 0$
$1000 - x > 0 \Longrightarrow x < 1000$
$2x - 1000 > 0 \Longrightarrow x > 500$
$2000 - 3x > 0 \Longrightarrow x < 666.\overline{6}$
$5x - 3000 > 0 \Longrightarrow x > 600$

And in general,

$n \equiv 0 \pmod{2}\quad\quad x < \frac{F_{n-1}}{F_n} \cdot 1000$
$n \equiv 1 \pmod{2}\quad\quad x > \frac{F_{n-1}}{F_{n}} \cdot 1000$

It is apparent that the bounds are slowly closing in on $x$ , so we can just calculate $x$ for some large value of $n$ (randomly, 10, 11):

$x < \frac{F_{7}}{F_{8}} \cdot 1000 = \frac{13}{21} \cdot 1000 = 618.\overline{18}$

$x > \frac{F_{8}}{F_{9}} \cdot 1000 = \frac{21}{34} \cdot 1000 \approx 617.977$

Thus the sequence is maximized when $x = \boxed{618}.$

Solution 2

It is well known that $\lim_{n\rightarrow\infty} \frac{F_{n-1}}{F_n} = \phi - 1 =\frac{1 + \sqrt{5}}{2} - 1 \approx .61803$ , so $1000 \cdot \frac{F_{n-1}}{F_n}$ approaches $x = \boxed{618}.$

@@ Line 2: / Line 2: @@
 Except for the first two terms, each term of the sequence <math>1000, x, 1000 - x,\ldots</math> is obtained by subtracting the preceding term from the one before that.  The last term of the sequence is the first [[negative]] term encounted.  What positive integer <math>x</math> produces a sequence of maximum length?
+__TOC__
+== Solution ==
+The best way to start is to just write out some terms.
+{| class="wikitable"
+|-
+| 0 || 1 || 2 || 3 || 4 || 5 || 6
+|-
+| <math>\quad 1000 \quad</math><font color="white">aa</font> || <math>\quad x \quad</math><font color="white">aaa</font> || <math>1000 - x</math> || <math>2x - 1000</math><font color="white">a</font> || <math>2000 - 3x</math> || <math>5x - 3000</math> || <math>5000 - 8x</math>
+|}
+It is now apparent that each term can be written as
+*<math>n \equiv 0 \pmod{2}\quad\quad F_{n-1}\cdot 1000-F_n\cdot x</math>
+*<math>n \equiv 1 \pmod{2}\quad\quad F_{n}\cdot 1000-F_{n-1}\cdot x</math>
+where the <math> F_{n}</math> are [[Fibonacci number]]s. This can be proven through induction.
+=== Solution 1 ===
+We can start to write out some of the inequalities now:
+#<math>x > 0</math>
+#<math>1000 - x > 0 \Longrightarrow x < 1000</math>
+#<math>2x - 1000 > 0 \Longrightarrow x > 500</math>
+#<math>2000 - 3x > 0 \Longrightarrow x < 666.\overline{6}</math>
+#<math>5x - 3000 > 0 \Longrightarrow x > 600</math>
-Solutions were removed
+And in general,
-__TOC__
+*<math>n \equiv 0 \pmod{2}\quad\quad x < \frac{F_{n-1}}{F_n} \cdot 1000</math>
+*<math>n \equiv 1 \pmod{2}\quad\quad x > \frac{F_{n-1}}{F_{n}} \cdot 1000</math>
+It is apparent that the bounds are slowly closing in on <math>x</math>, so we can just calculate <math>x</math> for some large value of <math>n</math> (randomly, 10, 11):
+<math>x < \frac{F_{7}}{F_{8}} \cdot 1000 = \frac{13}{21} \cdot 1000 = 618.\overline{18}</math>
+<math>x > \frac{F_{8}}{F_{9}} \cdot 1000 = \frac{21}{34} \cdot 1000 \approx 617.977</math>
+Thus the sequence is maximized when <math>x = \boxed{618}.</math>
+=== Solution 2 ===
+It is well known that <math>\lim_{n\rightarrow\infty} \frac{F_{n-1}}{F_n} = \phi - 1 =\frac{1 + \sqrt{5}}{2} - 1 \approx .61803</math>, so <math>1000 \cdot \frac{F_{n-1}}{F_n}</math> approaches <math>x = \boxed{618}.</math>
 == See also ==

1998 AIME (Problems • Answer Key • Resources)
Preceded by Problem 7	Followed by Problem 9
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