Difference between revisions of "2002 AMC 10B Problems/Problem 23"
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== Solution 2 == | == Solution 2 == | ||
− | Substituting <math>n=1</math> into <math>a_{m+n}=a_m+a_n+mn</math>: <math>a_{m+1}=a_m+a_{1}+m</math>. Since <math>a_1 = 1</math>, <math>a_{m+1}=a_m+m+1</math>. Therefore, <math>a_m = a_{m-1} + m, a_{m-1}=a_{m-2}+(m-1), a_{m-2} = a_{m-3} + (m-2)</math>, and so on until <math>a_2 = a_1 + 2</math>. Adding the Left Hand Sides of all of these equations gives <math>a_m + a_{m-1} + a_{m-2} + a_{m-3} + \cdots + a_2</math> | + | Substituting <math>n=1</math> into <math>a_{m+n}=a_m+a_n+mn</math>: <math>a_{m+1}=a_m+a_{1}+m</math>. |
+ | |||
+ | Since <math>a_1 = 1</math>, <math>a_{m+1}=a_m+m+1</math>. | ||
+ | |||
+ | Therefore, <math>a_m = a_{m-1} + m, a_{m-1}=a_{m-2}+(m-1), a_{m-2} = a_{m-3} + (m-2)</math>, and so on until <math>a_2 = a_1 + 2</math>. | ||
+ | |||
+ | Adding the Left Hand Sides of all of these equations gives <math>a_m + a_{m-1} + a_{m-2} + a_{m-3} + \cdots + a_2</math>. | ||
+ | |||
+ | Adding the Right Hand Sides of these equations gives | ||
+ | |||
+ | <math>(a_{m-1} + a_{m-2} + a_{m-3} + \cdots + a_1) + (m + (m-1) + (m-2) + \cdots + 2)</math>. | ||
+ | |||
+ | These two expressions must be equal; hence <math>a_m + a_{m-1} + a_{m-2} + a_{m-3} + \cdots + a_2 = (a_{m-1} + a_{m-2} + a_{m-3} + \cdots + a_1) + (m + (m-1) + (m-2) + \cdots + 2)</math> and <math>a_m = a_1 + (m + (m-1) + (m-2) + \cdots + 2)</math>. | ||
+ | |||
+ | Substituting <math>a_1 = 1</math>: <math>a_m = 1 + (m + (m-1) + (m-2) + \cdots + 2) = 1+2+3+4+ \cdots +m = \frac{(m+1)(m)}{2}</math>. | ||
+ | |||
+ | Thus we have a general formula for <math>a_m</math> and substituting <math>m=12</math>: <math>a_{12} = \frac{(13)(12)}{2} = (13)(6) = \boxed{\mathrm{(D)} 78}</math>. | ||
==Solution 3== | ==Solution 3== | ||
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</asy> | </asy> | ||
− | This means that we can use the triangular number formula <math>T_n = \frac{n(n+1)}{2}</math>, so the answer is <math>T_{12} = \frac{12(12+1)}{2} = \boxed{\mathrm{(D) \ } 78}</math>. ~[[User:emerald_block|emerald_block]] | + | This means that we can use the triangular number formula <math>T_n = \frac{n(n+1)}{2}</math>, so the answer is <math>T_{12} = \frac{12(12+1)}{2} = \boxed{\mathrm{(D) \ } 78}</math>. |
+ | |||
+ | |||
+ | ~[[User:emerald_block|emerald_block]] | ||
+ | |||
+ | |||
+ | == Solution 6 == | ||
+ | Follow solution 4 with plugging in m=n, but for the last part since since the subscript is 2n, and we aren't given anything that has odd numbers ( without substitution into the original equation), we need to find two numbers that add up to 12, which result from the multiplication of two even numbers. We realize that 4+8 works, since 4=2x2 and 8=4x2. We then proceed in substituting n=2 and n=4 into the equation, then add to obtain 78. | ||
+ | ~Charmainema07292010 | ||
+ | |||
+ | == Video Solution == | ||
+ | https://www.youtube.com/watch?v=zraGzYAh0uM ~David | ||
== See also == | == See also == | ||
{{AMC10 box|year=2002|ab=B|num-b=22|num-a=24}} | {{AMC10 box|year=2002|ab=B|num-b=22|num-a=24}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 06:38, 23 October 2023
Contents
Problem 23
Let be a sequence of integers such that
and
for all positive integers
and
Then
is
Solution 1
When ,
. Hence,
Adding these equations up, we have that
~AopsUser101
Solution 2
Substituting into
:
.
Since ,
.
Therefore, , and so on until
.
Adding the Left Hand Sides of all of these equations gives .
Adding the Right Hand Sides of these equations gives
.
These two expressions must be equal; hence and
.
Substituting :
.
Thus we have a general formula for and substituting
:
.
Solution 3
We can literally just plug stuff in. No prerequisite is actually said in the sequence. Since , we know
. After this, we can use
to find
.
. Now, we can use
and
to find
, or
. Lastly, we can use
to find
.
Solution 4
We can set equal to
, so we can say that
We set , we get
.
We set m, we get
.
Solving for is easy, just direct substitution.
Substituting, we get
Thus, the answer is .
~ euler123
Solution 5
Note that the sequence of triangular numbers satisfies these conditions. It is immediately obvious that it satisfies
, and
can be visually proven with the diagram below.
This means that we can use the triangular number formula , so the answer is
.
Solution 6
Follow solution 4 with plugging in m=n, but for the last part since since the subscript is 2n, and we aren't given anything that has odd numbers ( without substitution into the original equation), we need to find two numbers that add up to 12, which result from the multiplication of two even numbers. We realize that 4+8 works, since 4=2x2 and 8=4x2. We then proceed in substituting n=2 and n=4 into the equation, then add to obtain 78. ~Charmainema07292010
Video Solution
https://www.youtube.com/watch?v=zraGzYAh0uM ~David
See also
2002 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 22 |
Followed by Problem 24 | |
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 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.