Difference between revisions of "2018 AMC 12B Problems/Problem 9"
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<cmath> \sum^{100}_{i=1} \sum^{100}_{j=1} (i+j) ? </cmath> | <cmath> \sum^{100}_{i=1} \sum^{100}_{j=1} (i+j) ? </cmath> | ||
− | <math> \textbf{(A) }100,100 \qquad | + | <math> \textbf{(A) }100{,}100 \qquad |
− | \textbf{(B) }500,500\qquad | + | \textbf{(B) }500{,}500\qquad |
− | \textbf{(C) }505,000 \qquad | + | \textbf{(C) }505{,}000 \qquad |
− | \textbf{(D) }1,001,000 \qquad | + | \textbf{(D) }1{,}001{,}000 \qquad |
− | \textbf{(E) }1,010,000 \qquad </math> | + | \textbf{(E) }1{,}010{,}000 \qquad </math> |
== Solution 1 == | == Solution 1 == | ||
We can start by writing out the first couple of terms: | We can start by writing out the first couple of terms: | ||
− | + | <cmath>\begin{array}{ccccccccc} | |
− | <cmath>(1+1) + (1+2) + (1+3) + \dots + (1+100) | + | (1+1) &+ &(1+2) &+ &(1+3) &+ &\dots &+ &(1+100) \\ |
− | + | (2+1) &+ &(2+2) &+ &(2+3) &+ &\dots &+ &(2+100) \\ | |
− | + | (3+1) &+ &(3+2) &+ &(3+3) &+ &\dots &+ &(3+100) \\ [-1ex] | |
− | + | &&&&\vdots&&&& \\ | |
− | + | (100+1) &+ &(100+2) &+ &(100+3) &+ &\dots &+ &(100+100) | |
− | + | \end{array}</cmath> | |
+ | Looking at the first terms in the parentheses, we can see that <math>1+2+3+\dots+100</math> occurs <math>100</math> times. It goes vertically and exists <math>100</math> times horizontally. | ||
Looking at the second terms in the parentheses, we can see that <math>1+2+3+\dots+100</math> occurs <math>100</math> times. It goes horizontally and exists <math>100</math> times vertically. | Looking at the second terms in the parentheses, we can see that <math>1+2+3+\dots+100</math> occurs <math>100</math> times. It goes horizontally and exists <math>100</math> times vertically. | ||
− | |||
− | Thus, we have | + | Thus, we have <cmath>2\left(\dfrac{100\cdot101}{2}\cdot 100\right)=\boxed{\textbf{(E) }1{,}010{,}000}.</cmath> |
− | |||
− | |||
== Solution 2 == | == Solution 2 == | ||
− | + | We have | |
− | <cmath> \sum^{100}_{i=1} \sum^{100}_{j=1} (i+j) = \sum^{100}_{i=1} (100i+5050) = 100 \cdot 5050 + 5050 \cdot 100 = \boxed{1,010,000} </cmath> | + | <cmath> \sum^{100}_{i=1} \sum^{100}_{j=1} (i+j) = \sum^{100}_{i=1} (100i+5050) = 100 \cdot 5050 + 5050 \cdot 100 = \boxed{\textbf{(E) }1{,}010{,}000}.</cmath> |
== Solution 3 == | == Solution 3 == | ||
− | + | We have | |
− | <cmath> \sum^{100}_{i=1} \sum^{100}_{j=1} (i+j) = \sum^{100}_{i=1} \sum^{100}_{i=1} 2i = | + | <cmath> \sum^{100}_{i=1} \sum^{100}_{j=1} (i+j) = \sum^{100}_{i=1} \sum^{100}_{i=1} 2i = 100\cdot(5050\cdot2) = \boxed{\textbf{(E) }1{,}010{,}000}.</cmath> |
== Solution 4 == | == Solution 4 == | ||
− | The minimum term is <math>1 + 1 = 2</math>, and the maximum term is <math>100 + 100 = 200</math>. The average of the <math>100 \cdot 100 = 10,000</math> terms is the average of the minimum and maximum terms, which is <math>\frac{2+200}{2}=101</math>. The sum is therefore <math>101 \cdot 10,000 = \boxed{1,010,000}</math> | + | The minimum term is <math>1 + 1 = 2</math>, and the maximum term is <math>100 + 100 = 200</math>. The average of the <math>100 \cdot 100 = 10{,}000</math> terms is the average of the minimum and maximum terms, which is <math>\frac{2+200}{2}=101</math>. The sum is therefore <math>101 \cdot 10{,}000 = \boxed{\textbf{(E) }1{,}010{,}000}</math>. |
==See Also== | ==See Also== |
Revision as of 11:25, 20 September 2021
Problem
What is
Solution 1
We can start by writing out the first couple of terms: Looking at the first terms in the parentheses, we can see that occurs times. It goes vertically and exists times horizontally. Looking at the second terms in the parentheses, we can see that occurs times. It goes horizontally and exists times vertically.
Thus, we have
Solution 2
We have
Solution 3
We have
Solution 4
The minimum term is , and the maximum term is . The average of the terms is the average of the minimum and maximum terms, which is . The sum is therefore .
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 |
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