Difference between revisions of "2018 AMC 12B Problems/Problem 9"
MRENTHUSIASM (talk | contribs) (Rearranged the solutions based on similar ideas.) |
MRENTHUSIASM (talk | contribs) (→Solution 5: Made this solution more concise and professional.) |
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== Solution 5 == | == Solution 5 == | ||
− | We can start by writing out the first | + | We can start by writing out the first few terms: |
<cmath>\begin{array}{ccccccccc} | <cmath>\begin{array}{ccccccccc} | ||
− | (1+1) &+ &(1+2) &+ &(1+3) &+ &\ | + | (1+1) &+ &(1+2) &+ &(1+3) &+ &\cdots &+ &(1+100) \\ |
− | (2+1) &+ &(2+2) &+ &(2+3) &+ &\ | + | (2+1) &+ &(2+2) &+ &(2+3) &+ &\cdots &+ &(2+100) \\ |
− | (3+1) &+ &(3+2) &+ &(3+3) &+ &\ | + | (3+1) &+ &(3+2) &+ &(3+3) &+ &\cdots &+ &(3+100) \\ [-1ex] |
&&&&\vdots&&&& \\ | &&&&\vdots&&&& \\ | ||
− | (100+1) &+ &(100+2) &+ &(100+3) &+ &\dots &+ &(100+100) | + | (100+1) &+ &(100+2) &+ &(100+3) &+ &\dots &+ &(100+100). |
\end{array}</cmath> | \end{array}</cmath> | ||
− | + | From the first terms in the parentheses, the sum <math>1+2+3+\dots+100</math> occurs <math>100</math> times vertically. | |
− | |||
− | + | From the second terms in the parentheses, the sum <math>1+2+3+\dots+100</math> occurs <math>100</math> times horizontally. | |
+ | |||
+ | Recall that the sum of the first <math>100</math> positive integers is <math>1+2+3+\cdots+100=\frac{101\cdot100}{2}=5050.</math> Therefore, the answer is <cmath>2\left(5050\cdot100\right)=\boxed{\textbf{(E) }1{,}010{,}000}.</cmath> | ||
+ | ~RandomPieKevin ~MRENTHUSIASM | ||
== Solution 6 == | == Solution 6 == |
Revision as of 14:51, 20 September 2021
Problem
What is
Solution 2
Recall that the sum of the first positive integers is It follows that ~Vfire ~MRENTHUSIASM
Solution 3
Recall that the sum of the first positive integers is Since the nested summation is symmetric with respect to and it follows that ~Vfire ~MRENTHUSIASM
Solution 4
The sum contains terms, and the average value of both and is Therefore, the sum becomes ~Rejas ~MRENTHUSIASM
Solution 5
We can start by writing out the first few terms: From the first terms in the parentheses, the sum occurs times vertically.
From the second terms in the parentheses, the sum occurs times horizontally.
Recall that the sum of the first positive integers is Therefore, the answer is ~RandomPieKevin ~MRENTHUSIASM
Solution 6
When we expand the nested summation as shown in Solution 5, note that:
- The term appears time.
The term appears times.
The term appears times.
The term appears times.
More generally, the term appears times for
- The term appears times.
The term appears times.
The term appears times.
The term appears time.
More generally, the term appears times for
Together, the nested summation becomes ~MRENTHUSIASM
See Also
2018 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 8 |
Followed by Problem 10 |
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 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.