The houses on one side of a road are numbered | vedclass

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Question

(C) Let the number of houses be $x, x+2, x+4, x+6, x+8, x+10, \dots$ where $x$ is the first house number.Since $a$ is the $6^{th}$ house number,$a = x

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$14 \leq a \leq 20$

Vedclass · Answer

(C) Let the number of houses be $x, x+2, x+4, x+6, x+8, x+10, \dots$ where $x$ is the first house number.Since $a$ is the $6^{th}$ house number,$a = x + 10$,which implies $x = a - 10$.Since house numbers must be positive even integers,$x \geq 2$,so $a - 10 \geq 2 \Rightarrow a \geq 12$.The sum of $n$ houses is given by $S_n = \frac{n}{2}[2x + (n-1)2] = n(x + n - 1) = 170$.Substituting $x = a - 10$,we get $n(a - 10 + n - 1) = 170$,or $n(a + n - 11) = 170$.Since $n \geq 6$,we test factors of $170$ $(1, 2, 5, 10, 17, 34, 85, 170)$.If $n=10$,$10(a + 10 - 11) = 170$ $\Rightarrow a - 1 = 17$ $\Rightarrow a = 18$.If $n=5$ (not allowed as $n \geq 6$),$5(a + 5 - 11) = 170$ $\Rightarrow a - 6 = 34$ $\Rightarrow a = 40$.If $n=17$,$17(a + 17 - 11) = 170$ $\Rightarrow a + 6 = 10$ $\Rightarrow a = 4$.Since $a \geq 12$,the valid range for $a$ containing $18$ is $14 \leq a \leq 20$.

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:

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