Difference between revisions of "1989 AIME Problems/Problem 7"
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If the integer <math>k</math> is added to each of the numbers <math>36</math>, <math>300</math>, and <math>596</math>, one obtains the squares of three consecutive terms of an arithmetic series. Find <math>k</math>. | If the integer <math>k</math> is added to each of the numbers <math>36</math>, <math>300</math>, and <math>596</math>, one obtains the squares of three consecutive terms of an arithmetic series. Find <math>k</math>. | ||
− | == Solution == | + | == Solution 1== |
Call the terms of the [[arithmetic progression]] <math>a,\ a + d,\ a + 2d</math>, making their squares <math>a^2,\ a^2 + 2ad + d^2,\ a^2 + 4ad + 4d^2</math>. | Call the terms of the [[arithmetic progression]] <math>a,\ a + d,\ a + 2d</math>, making their squares <math>a^2,\ a^2 + 2ad + d^2,\ a^2 + 4ad + 4d^2</math>. | ||
Line 8: | Line 8: | ||
Subtracting the first equation from the second, we get <math>2d^2 = 32</math>, so <math>d = 4</math>. Substituting backwards yields that <math>a = 31</math> and <math>k = \boxed{925}</math>. | Subtracting the first equation from the second, we get <math>2d^2 = 32</math>, so <math>d = 4</math>. Substituting backwards yields that <math>a = 31</math> and <math>k = \boxed{925}</math>. | ||
+ | |||
+ | ==Solution 2 (Straighforward, but has big numbers)== | ||
+ | Since terms in an arithmetic progression have constant differences, | ||
+ | <cmath>\sqrt{300+k}-\sqrt{36+k}=\sqrt{596+k}-\sqrt{300+k}</cmath> | ||
+ | <cmath>\implies 2\sqrt{300+k} = \sqrt{596+k}+\sqrt{36+k}</cmath> | ||
+ | <cmath>\implies 4k+1200=596+k+36+k+2\sqrt{(596+k)(36+k)}</cmath> | ||
+ | <cmath>\implies 2k+568=2\sqrt{(596+k)(36+k)}</cmath> | ||
+ | <cmath>\implies k+284=\sqrt{(596+k)(36+k)}</cmath> | ||
+ | <cmath>\implies k^2+568k+80656=k^2+632k+21456</cmath> | ||
+ | <cmath>\implies 568k+80656=632k+21456</cmath> | ||
+ | <cmath>\implies 64k = 59200</cmath> | ||
+ | <cmath>\implies k = \boxed{925}</cmath> | ||
+ | |||
+ | == Solution 3 == | ||
+ | Let the arithmetic sequence be <math>a-d</math>, <math>a</math>, and <math>a+d</math>. Then <math>(a+d)^2-a^2 = 296</math>, but using the difference of squares, <math>d(2a+d)=296</math>. Also, <math>a^2-(a-d)^2 = 264</math>, and using the difference of squares we get <math>d(2a-d) = 264</math>. Subtracting both equations gives <math>4d^2 = 32</math>, <math>d = 4</math>, and <math>a = 35</math>. Since <math>a = 35</math>, <math>a^2 = 1225 = 300+k</math> and <math>k = \boxed{925}</math>. | ||
+ | |||
+ | ~~Disphenoid_lover | ||
+ | |||
+ | == Video Solution by OmegaLearn == | ||
+ | https://youtu.be/qL0OOYZiaqA?t=251 | ||
+ | |||
+ | ~ pi_is_3.14 | ||
== See also == | == See also == | ||
{{AIME box|year=1989|num-b=6|num-a=8}} | {{AIME box|year=1989|num-b=6|num-a=8}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
+ | [[Category: Introductory Algebra Problems]] |
Latest revision as of 15:21, 14 May 2024
Contents
Problem
If the integer is added to each of the numbers
,
, and
, one obtains the squares of three consecutive terms of an arithmetic series. Find
.
Solution 1
Call the terms of the arithmetic progression , making their squares
.
We know that and
, and subtracting these two we get
(1). Similarly, using
and
, subtraction yields
(2).
Subtracting the first equation from the second, we get , so
. Substituting backwards yields that
and
.
Solution 2 (Straighforward, but has big numbers)
Since terms in an arithmetic progression have constant differences,
Solution 3
Let the arithmetic sequence be ,
, and
. Then
, but using the difference of squares,
. Also,
, and using the difference of squares we get
. Subtracting both equations gives
,
, and
. Since
,
and
.
~~Disphenoid_lover
Video Solution by OmegaLearn
https://youtu.be/qL0OOYZiaqA?t=251
~ pi_is_3.14
See also
1989 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
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.