# 2018 AMC 12B Problems/Problem 9

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## 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

We can start by writing out the first couple of terms:

$$(1+1) + (1+2) + (1+3) + \dots + (1+100)$$ $$(2+1) + (2+2) + (2+3) + \dots + (2+100)$$ $$(3+1) + (3+2) + (3+3) + \dots + (3+100)$$ $$\vdots$$ $$(100+1) + (100+2) + (100+3) + \dots + (100+100)$$

Looking at the second terms in the parentheses, we can see that $1+2+3+\dots+100$ occurs $100$ times. It goes horizontally and exists $100$ times vertically. Looking at the first terms in the parentheses, we can see that $1+2+3+\dots+100$ occurs $100$ times. It goes vertically and exists $100$ times horizontally.

Thus, we have: $$2\left(\dfrac{100\cdot101}{2}\cdot 100\right).$$

This gives us: $$\boxed{\textbf{(E) } 1010000}.$$

## Solution 2

$$\sum^{100}_{i=1} \sum^{100}_{j=1} (i+j) = \sum^{100}_{i=1} (100i+5050) = 100 \cdot 5050 + 5050 \cdot 100 = \boxed{1,010,000}$$

## Solution 3

$$\sum^{100}_{i=1} \sum^{100}_{j=1} (i+j) = \sum^{100}_{i=1} \sum^{100}_{i=1} 2i = (100)*(5050*2) = \boxed{1,010,000}$$

## Solution 4

The minimum term is $1 + 1 = 2$, and the maximum term is $100 + 100 = 200$. The average of the $100 \cdot 100 = 10,000$ terms is the average of the minimum and maximum terms, which is $\frac{2+200}{2}=101$. The sum is therefore $101 \cdot 10,000 = \boxed{1,010,000}$