Question

¿Cuál es el valor de $\begin{align*} &1+2+3-4 +5+6+7-8\\ &+\cdots+197+198+199-200 \end{align*}$ ?

What is the value of $\begin{align*} &1+2+3-4 +5+6+7-8\\ &+\cdots+197+198+199-200? \end{align*}$

$9,800$

$9,900$

$10,000$

$10,100$

$10,200$

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Agrupa en bloques de cuatro: $(1+2+3-4)$ $+(5+6+7-8)$ $+\cdots$ $+(197+198+199-200)$ . El bloque $j$ suma $(4j-3)+(4j-2)$ $+(4j-1)-4j=8j-6$ .

Hay $50$ bloques, así que la suma es $\sum_{j=1}^{50}(8j-6)=8\cdot\dfrac{50\cdot51}{2}$ $-6\cdot50=9900$ . Así, B es la respuesta correcta.

Group the terms in blocks of four: $(1+2+3-4)$ $+(5+6+7-8)$ $+\cdots$ $+(197+198+199-200)$ . The $j$ th block is $(4j-3)+(4j-2)$ $+(4j-1)-4j=8j-6$ .

There are $50$ blocks, so the sum is $\sum_{j=1}^{50}(8j-6)=8\cdot\dfrac{50\cdot51}{2}$ $-6\cdot50=9900$ . Thus, B is the correct answer.