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A multistorey building is constructed with stilt parking. There is a provision of lift, as well as staircase from the ground floor to the top floor. The number of stairs from the ground floor to the first floor is $10$, from the first floor to the second floor is $24$, from the second floor to the third floor is $38$, and so on.
Based on the above information, answer the following questions :
(i) Does $10, 24, 38, ...$ form an A.P.? Justify your answer.
(ii) What will be the total number of stairs from the ground floor to the eleventh floor?
(iii) A person supplies water cans to people living in the building. As water cans are heavy, he supplies water cans on each floor, carrying one at a time. He supplied the water can from the ground floor to the first floor, came back and supplied water can to the second floor, again came back then supplied water can to the third floor, and so on.
(a) Find the total number of stairs he climbed up and down to supply water till the sixth floor, using A.P.
OR
(b) The next day, following the same process, if a person climbed up and down a total of $380$ stairs, till which floor did he supply water cans ?
Based on the above information, answer the following questions :
(i) Does $10, 24, 38, ...$ form an A.P.? Justify your answer.
(ii) What will be the total number of stairs from the ground floor to the eleventh floor?
(iii) A person supplies water cans to people living in the building. As water cans are heavy, he supplies water cans on each floor, carrying one at a time. He supplied the water can from the ground floor to the first floor, came back and supplied water can to the second floor, again came back then supplied water can to the third floor, and so on.
(a) Find the total number of stairs he climbed up and down to supply water till the sixth floor, using A.P.
OR
(b) The next day, following the same process, if a person climbed up and down a total of $380$ stairs, till which floor did he supply water cans ?
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Solution:
(i) $10, 24, 38, ...$
$24-10 = 14$ and $38 - 24 = 14$, so $d = 14$ (1/2 Mark)
Hence, $10, 24, 38, ...$ form an A. P. (1/2 Mark)
(ii) $S_{11} = \frac{11}{2} (2 \times 10 + 10 \times 14) = 880$ stairs (1 Mark)
(iii) (a) Number of stairs to reach $n^{th}$ floor $= 7n^2+3n$ (1 Mark)
$:::$ Total number of stairs he climbed up and down to supply water till the sixth floor
$= 2S_1+ 2S_2+ 2S_3+ 2S_4+ 2S_5+ 2S_6 = 20+68+144+248 +380+540=1400$ (1 Mark)
OR
(iii) (b) As $20+68+144+248 > 380$ (1 Mark)
He supplied water till $3^{rd}$ floor (1 Mark)
(i) $10, 24, 38, ...$
$24-10 = 14$ and $38 - 24 = 14$, so $d = 14$ (1/2 Mark)
Hence, $10, 24, 38, ...$ form an A. P. (1/2 Mark)
(ii) $S_{11} = \frac{11}{2} (2 \times 10 + 10 \times 14) = 880$ stairs (1 Mark)
(iii) (a) Number of stairs to reach $n^{th}$ floor $= 7n^2+3n$ (1 Mark)
$:::$ Total number of stairs he climbed up and down to supply water till the sixth floor
$= 2S_1+ 2S_2+ 2S_3+ 2S_4+ 2S_5+ 2S_6 = 20+68+144+248 +380+540=1400$ (1 Mark)
OR
(iii) (b) As $20+68+144+248 > 380$ (1 Mark)
He supplied water till $3^{rd}$ floor (1 Mark)