Question

The houses on one side of a road are numbered using consecutive even numbers. The sum of the numbers of all the houses in that row is $170$. If there are at least $6$ houses in that row and $a$ is the number of the sixth house, then

• A. $2\le a\le 6$
• B. $8\le a\le 12$
• C. $14\le a\le 20$
• D. $22\le a\le 30$

Answer 1

Answer: (C)

Let the number of houses be
$x,x+2,x+4,x+6,x+8,x+10,...$
$6$th number of house is $a$.
$\because x+10=a$
$\Rightarrow x=a-10$
$\therefore x>10$
Now, $S_n=\frac{n}{2}(2x+(n-1)2)$
$S_n=n(x+n-1)$
$\Rightarrow 170=n(a-10+n-1)$
$\Rightarrow n^2+(a-11)n-170=0$
$\Rightarrow n=-\frac{\left(a-11\right)\pm \sqrt{{\left(a-11\right)}^2+680}}{2}$
$\Rightarrow n=\frac{\left(11-a\right)\pm \sqrt{{\left(a-11\right)}^2+680}}{2}n\ge 6$
$\because \frac{\left(11-a\right)\pm \sqrt{{\left(a-11\right)}^2+680}}{2}\ge 6$
$\Rightarrow a\le \frac{800}{24}\le 33.33$
$\because 12\le a\le 32$
$a=12,14,16,18,...$
When, $a=18,n=10$, then $S_n=170$
$\because a=18$