Question

¿Cuánto vale $\sum_{i=1}^{100}\sum_{j=1}^{100}(i+j)?$

What is $\sum_{i=1}^{100}\sum_{j=1}^{100}(i+j)?$

$100{,}100$

$500{,}500$

$505{,}000$

$1{,}001{,}000$

$1{,}010{,}000$

Solución:

Separando la suma, $\begin{gathered} \sum_{i=1}^{100}\sum_{j=1}^{100}(i+j) \\ =\sum_{i=1}^{100}\sum_{j=1}^{100}i \\ {}+\sum_{i=1}^{100}\sum_{j=1}^{100}j \\ =100\sum_{i=1}^{100}i \\ {}+100\sum_{j=1}^{100}j. \end{gathered}$

Como $\sum_{k=1}^{100}k=5050,$ esto es igual a $100\cdot5050$ $+100\cdot5050$ $=1{,}010{,}000.$

Por lo tanto, la respuesta correcta es E.

Splitting the sum, $\begin{gathered} \sum_{i=1}^{100}\sum_{j=1}^{100}(i+j) \\ =\sum_{i=1}^{100}\sum_{j=1}^{100}i \\ {}+\sum_{i=1}^{100}\sum_{j=1}^{100}j \\ =100\sum_{i=1}^{100}i \\ {}+100\sum_{j=1}^{100}j. \end{gathered}$

Since $\sum_{k=1}^{100}k=5050,$ this equals $100\cdot5050$ $+100\cdot5050$ $=1{,}010{,}000.$

Thus, the correct answer is E.

2018 AMC 12B Problema 9

Solución:

El Problema 9 en otros años