Difference between revisions of "2018 AMC 12B Problems/Problem 9"

m (Solution 5)
 
(25 intermediate revisions by 7 users not shown)
Line 4: Line 4:
 
<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 ==
 +
Recall that the sum of the first <math>100</math> positive integers is <math>\sum^{100}_{k=1} k = \frac{101\cdot100}{2}=5050.</math> It follows that
 +
<cmath>\begin{align*}
 +
\sum^{100}_{i=1} \sum^{100}_{j=1} (i+j) &= \sum^{100}_{i=1} \sum^{100}_{j=1}i + \sum^{100}_{i=1} \sum^{100}_{j=1}j \\
 +
&= \sum^{100}_{i=1} (100i) + 100 \sum^{100}_{j=1}j \\
 +
&= 100 \sum^{100}_{i=1}i + 100 \sum^{100}_{j=1}j \\
 +
&= 100\cdot5050 + 100\cdot5050 \\
 +
&= \boxed{\textbf{(E) }1{,}010{,}000}.
 +
\end{align*}</cmath>
 +
~MRENTHUSIASM
  
<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> (Vfire)
+
== Solution 2 ==
 +
Recall that the sum of the first <math>100</math> positive integers is <math>\sum^{100}_{k=1} k = \frac{101\cdot100}{2}=5050.</math> It follows that
 +
<cmath>\begin{align*}
 +
\sum^{100}_{i=1} \sum^{100}_{j=1} (i+j) &= \sum^{100}_{i=1} \left(\sum^{100}_{j=1} i + \sum^{100}_{j=1} j\right) \\
 +
&= \sum^{100}_{i=1} (100i+5050) \\
 +
&= 100\sum^{100}_{i=1} i + \sum^{100}_{i=1} 5050 \\
 +
&= 100\cdot5050+5050\cdot100 \\
 +
&= \boxed{\textbf{(E) }1{,}010{,}000}.
 +
\end{align*}</cmath>
 +
~Vfire ~MRENTHUSIASM
  
 +
== Solution 3 ==
 +
Recall that the sum of the first <math>100</math> positive integers is <math>\sum^{100}_{k=1} k = \frac{101\cdot100}{2}=5050.</math> Since the nested summation is symmetric with respect to <math>i</math> and <math>j,</math> it follows that
 +
<cmath>\begin{align*}
 +
\sum^{100}_{i=1} \sum^{100}_{j=1} (i+j) &= \sum^{100}_{i=1} \sum^{100}_{i=1} (2i) \\
 +
&= 2\sum^{100}_{i=1} \sum^{100}_{i=1} i \\
 +
&= 2\sum^{100}_{i=1} 5050 \\
 +
&= 2\cdot(5050\cdot100) \\
 +
&= \boxed{\textbf{(E) }1{,}010{,}000}.
 +
\end{align*}</cmath>
 +
~Vfire ~MRENTHUSIASM
  
== Solution 2 ==
+
== Solution 4 ==
 +
The sum contains <math>100\cdot100=10000</math> terms, and the average value of both <math>i</math> and <math>j</math> is <math>\frac{101}{2}.</math> Therefore, the sum becomes <cmath>10000\cdot\left(\frac{101}{2}+\frac{101}{2}\right)=\boxed{\textbf{(E) }1{,}010{,}000}.</cmath>
 +
~Rejas ~MRENTHUSIASM
 +
 
 +
== Solution 5 ==
 +
 
 +
We start by writing out the first few terms:
 +
<cmath>\begin{array}{ccccccccc}
 +
(1+1) &+ &(1+2) &+ &(1+3) &+ &\cdots &+ &(1+100) \\
 +
(2+1) &+ &(2+2) &+ &(2+3) &+ &\cdots &+ &(2+100) \\
 +
(3+1) &+ &(3+2) &+ &(3+3) &+ &\cdots &+ &(3+100) \\ [-1ex]
 +
&&&&\vdots&&&& \\
 +
(100+1) &+ &(100+2) &+ &(100+3) &+ &\cdots &+ &(100+100).
 +
\end{array}</cmath>
 +
From the first terms in the parentheses, the sum <math>1+2+3+\cdots+100</math> occurs <math>100</math> times vertically.
 +
 
 +
From the second terms in the parentheses, the sum <math>1+2+3+\cdots+100</math> occurs <math>100</math> times horizontally.
  
<cmath> \sum^{100}_{i=1} \sum^{100}_{j=1} (i+j)  = \sum^{100}_{i=1} \sum^{100}_{i=1} 2i = (100)*(5050*2) = \boxed{1,010,000}   </cmath>
+
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\cdot\left(5050\cdot100\right)=\boxed{\textbf{(E) }1{,}010{,}000}.</cmath>
 +
~RandomPieKevin ‎~MRENTHUSIASM
  
 +
== Solution 6 ==
 +
When we expand the nested summation as shown in Solution 5, note that:
 +
<ol style="margin-left: 1.5em;">
 +
  <li>The term <math>2</math> occurs <math>1</math> time. <p>
 +
The term <math>3</math> occurs <math>2</math> times. <p>
 +
The term <math>4</math> occurs <math>3</math> times. <p>
 +
<math>\cdots</math> <p>
 +
The term <math>101</math> occurs <math>100</math> times. <p>
 +
More generally, the term <math>k+1</math> occurs <math>k</math> times for <math>k\in\{1,2,3,\ldots,100\}.</math><p></li>
 +
  <li>The term <math>102</math> occurs <math>99</math> times. <p>
 +
The term <math>103</math> occurs <math>98</math> times. <p>
 +
The term <math>104</math> occurs <math>97</math> times. <p>
 +
<math>\cdots</math> <p>
 +
The term <math>200</math> occurs <math>1</math> time. <p>
 +
More generally, the term <math>k+101</math> occurs <math>100-k</math> times for <math>k\in\{1,2,3,\ldots,99\}.</math><p></li>
 +
</ol>
 +
Together, the nested summation becomes
 +
<cmath>\begin{align*}
 +
\sum^{100}_{k=1}\left[(k+1)k\right] + \sum^{99}_{k=1}\left[(k+101)(100-k)\right] &= \sum^{100}_{k=1}\left[k^2+k\right] + \sum^{99}_{k=1}\left[-k^2-k+10100\right] \\
 +
&= \sum^{100}_{k=1}k^2 + \sum^{100}_{k=1}k - \sum^{99}_{k=1}k^2 - \sum^{99}_{k=1}k + \sum^{99}_{k=1}10100 \\
 +
&= \left(\sum^{100}_{k=1}k^2 - \sum^{99}_{k=1}k^2\right) + \left(\sum^{100}_{k=1}k - \sum^{99}_{k=1}k\right) + \sum^{99}_{k=1}10100 \\
 +
&= 100^2+100+10100\cdot99 \\
 +
&= \boxed{\textbf{(E) }1{,}010{,}000}.
 +
\end{align*}</cmath>
 +
~MRENTHUSIASM
  
 
==See Also==
 
==See Also==

Latest revision as of 12:26, 24 September 2021

Problem

What is \[\sum^{100}_{i=1} \sum^{100}_{j=1} (i+j) ?\]

$\textbf{(A) }100{,}100 \qquad \textbf{(B) }500{,}500\qquad \textbf{(C) }505{,}000 \qquad \textbf{(D) }1{,}001{,}000 \qquad \textbf{(E) }1{,}010{,}000 \qquad$

Solution 1

Recall that the sum of the first $100$ positive integers is $\sum^{100}_{k=1} k = \frac{101\cdot100}{2}=5050.$ It follows that \begin{align*} \sum^{100}_{i=1} \sum^{100}_{j=1} (i+j) &= \sum^{100}_{i=1} \sum^{100}_{j=1}i + \sum^{100}_{i=1} \sum^{100}_{j=1}j \\ &= \sum^{100}_{i=1} (100i) + 100 \sum^{100}_{j=1}j \\ &= 100 \sum^{100}_{i=1}i + 100 \sum^{100}_{j=1}j \\ &= 100\cdot5050 + 100\cdot5050 \\ &= \boxed{\textbf{(E) }1{,}010{,}000}. \end{align*} ~MRENTHUSIASM

Solution 2

Recall that the sum of the first $100$ positive integers is $\sum^{100}_{k=1} k = \frac{101\cdot100}{2}=5050.$ It follows that \begin{align*} \sum^{100}_{i=1} \sum^{100}_{j=1} (i+j) &= \sum^{100}_{i=1} \left(\sum^{100}_{j=1} i + \sum^{100}_{j=1} j\right) \\ &= \sum^{100}_{i=1} (100i+5050) \\ &= 100\sum^{100}_{i=1} i + \sum^{100}_{i=1} 5050 \\ &= 100\cdot5050+5050\cdot100 \\ &= \boxed{\textbf{(E) }1{,}010{,}000}. \end{align*} ~Vfire ~MRENTHUSIASM

Solution 3

Recall that the sum of the first $100$ positive integers is $\sum^{100}_{k=1} k = \frac{101\cdot100}{2}=5050.$ Since the nested summation is symmetric with respect to $i$ and $j,$ it follows that \begin{align*} \sum^{100}_{i=1} \sum^{100}_{j=1} (i+j) &= \sum^{100}_{i=1} \sum^{100}_{i=1} (2i) \\ &= 2\sum^{100}_{i=1} \sum^{100}_{i=1} i \\ &= 2\sum^{100}_{i=1} 5050 \\ &= 2\cdot(5050\cdot100) \\ &= \boxed{\textbf{(E) }1{,}010{,}000}. \end{align*} ~Vfire ~MRENTHUSIASM

Solution 4

The sum contains $100\cdot100=10000$ terms, and the average value of both $i$ and $j$ is $\frac{101}{2}.$ Therefore, the sum becomes \[10000\cdot\left(\frac{101}{2}+\frac{101}{2}\right)=\boxed{\textbf{(E) }1{,}010{,}000}.\] ~Rejas ~MRENTHUSIASM

Solution 5

We start by writing out the first few terms: \[\begin{array}{ccccccccc} (1+1) &+ &(1+2) &+ &(1+3) &+ &\cdots &+ &(1+100) \\ (2+1) &+ &(2+2) &+ &(2+3) &+ &\cdots &+ &(2+100) \\ (3+1) &+ &(3+2) &+ &(3+3) &+ &\cdots &+ &(3+100) \\ [-1ex] &&&&\vdots&&&& \\ (100+1) &+ &(100+2) &+ &(100+3) &+ &\cdots &+ &(100+100). \end{array}\] From the first terms in the parentheses, the sum $1+2+3+\cdots+100$ occurs $100$ times vertically.

From the second terms in the parentheses, the sum $1+2+3+\cdots+100$ occurs $100$ times horizontally.

Recall that the sum of the first $100$ positive integers is $1+2+3+\cdots+100=\frac{101\cdot100}{2}=5050.$ Therefore, the answer is \[2\cdot\left(5050\cdot100\right)=\boxed{\textbf{(E) }1{,}010{,}000}.\] ~RandomPieKevin ‎~MRENTHUSIASM

Solution 6

When we expand the nested summation as shown in Solution 5, note that:

  1. The term $2$ occurs $1$ time.

    The term $3$ occurs $2$ times.

    The term $4$ occurs $3$ times.

    $\cdots$

    The term $101$ occurs $100$ times.

    More generally, the term $k+1$ occurs $k$ times for $k\in\{1,2,3,\ldots,100\}.$

  2. The term $102$ occurs $99$ times.

    The term $103$ occurs $98$ times.

    The term $104$ occurs $97$ times.

    $\cdots$

    The term $200$ occurs $1$ time.

    More generally, the term $k+101$ occurs $100-k$ times for $k\in\{1,2,3,\ldots,99\}.$

Together, the nested summation becomes \begin{align*} \sum^{100}_{k=1}\left[(k+1)k\right] + \sum^{99}_{k=1}\left[(k+101)(100-k)\right] &= \sum^{100}_{k=1}\left[k^2+k\right] + \sum^{99}_{k=1}\left[-k^2-k+10100\right] \\ &= \sum^{100}_{k=1}k^2 + \sum^{100}_{k=1}k - \sum^{99}_{k=1}k^2 - \sum^{99}_{k=1}k + \sum^{99}_{k=1}10100 \\ &= \left(\sum^{100}_{k=1}k^2 - \sum^{99}_{k=1}k^2\right) + \left(\sum^{100}_{k=1}k - \sum^{99}_{k=1}k\right) + \sum^{99}_{k=1}10100 \\ &= 100^2+100+10100\cdot99 \\ &= \boxed{\textbf{(E) }1{,}010{,}000}. \end{align*} ~MRENTHUSIASM

See Also

2018 AMC 12B (ProblemsAnswer KeyResources)
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. AMC logo.png

Invalid username
Login to AoPS