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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

8 a 12

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B

14 a 20

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C

22 a 30

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D

2 a 6

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Solution

## The correct option is B 14 ≤ a ≤ 20 Let the house numbers be α,α+2,α+4,α+6,α+8,α+10….. α+10=a⇒α=a–10 …….(1) House no. will be (+) ⇒α=a–10>0 ⇒α=a–10>0 ⇒α>10 ⇒α≥ 12 as a is each too........... (2) Now,Sn=n2[2α+(n+1)d] 170=n2[2α+(n+1)2] =nα+(n+1)) =n(a−10+n−1) =n(a−11+n) ⇒n2+n(a−11)−170=0 ⇒n=(11−a)±√(a−11)2+6802 .....(3) ∵n≥6 ⇒(11−a)±√(a−11)2+6802≥6 ⇒a≤80024 .......(4) From (2) and (4) ⇒12≤a≤32 Now checking through (3) for a = 12, 14... we have a = 18, n = 10 and Sn = 170

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