Question

The house on one side of a road are numbered using consecutive even numbers. The sum of the numbers of all the house in that row is 170.If there are least 6 house 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)

$14\le a\le 20$

Let house no are $\alpha ,\alpha ,+2,\alpha +4,\alpha +6,\alpha +8,\alpha +10,........\alpha +=a\Rightarrow =\alpha =a-10$
House no.will be (+)
$\Rightarrow \alpha =a-10>0$
$\Rightarrow \alpha >10$
$\Rightarrow \alpha \ge 12$ as a is each too
Now $s_n=\frac{n}{2}[2\alpha +(n-1\left)d\right]$
$170=\frac{n}{2}[2\alpha +(n-1\left)\right(2\left)\right]$
$=n(\alpha +(n-1\left)\right)$
$=n(a-10+n-1)$
$=n(a-11+n)$
$\Rightarrow n^2=n(a-11)-170=0$
$n=\frac{(11-a)\pm \sqrt{(a-11{)}^2+680}}{2}\ge 6$
$\Rightarrow a\le \frac{800}{24}$
From (2) and (4) $Rightarrow12\le a\le 32$
now checking through (3) for a =12,14,......
we have a =18, n=10 and $S_n=170$
Hence option C is the answer