Problem Solving

Hello.

I liked this question and I want to try it. <i class="em em-grin"></i>

We have this (-20) + (-19) + (-18) + (-17) + . . . + (-2x - 4) = 90 and we want the value of x.

1. If x>0 then all the numbers in the sum are negative, hence their sum is negative, which is a contradiction because the sum is 90>0. Therefore, we can eliminate the options D and E.

2. Let's try option C, x=-10.

Then we have $$(-20)+(-19)+(-18)+(-17)+...+(-2\left(-10\right)-4)=(-20)+(-19)+(-18)+(-17)+...+(16)$$ $$=\left\{(-20)+(-19)+(-18)+\cdots+\left(-2\right)+\left(-1\right)\right\}+0+\left\{\left(1\right)+\left(2\right)+\cdots+\left(16\right)\right\}$$ $$=(-20)+(-19)+(-18)+\left(-17\right)$$ $$=\ -74\ \ne\ 90.$$ Therefore, we discard the option C.

3. Let's try option A, x=-24.

Then we have $$(-20)+(-19)+(-18)+(-17)+...+(-2\left(-24\right)-4)=(-20)+(-19)+(-18)+(-17)+...+(44)$$ $$=\left\{(-20)+(-19)+(-18)+\cdots+\left(-2\right)+\left(-1\right)\right\}+0+\left\{\left(1\right)+\left(2\right)+\cdots+\left(44\right)\right\}$$ $$=(21)+(22)+(23)++...+(44)$$ $$=\ 780\ \ne\ 90.$$ Therefore, we discard the option A.

4. Finally, let's try option B, x=-14.

Then we have $$(-20)+(-19)+(-18)+(-17)+...+(-2\left(-14\right)-4)=(-20)+(-19)+(-18)+(-17)+...+(24)$$ $$=\left\{(-20)+(-19)+(-18)+\cdots+\left(-2\right)+\left(-1\right)\right\}+0+\left\{\left(1\right)+\left(2\right)+\cdots+\left(24\right)\right\}$$ $$=(21)+(22)+(23)+(24)$$ $$=\ 90 = 90.$$ Therefore, the correct answer is the option [spoiler]B=-14[/spoiler].

I'd like to see a shorter way. <i class="em em-wink"></i>