216 logs are stacked in the following manner as 21 logs in the bottom row, 20 in the next row, 19 in the next row to it and so on. In how many rows are the 216 logs placed and how many logs are in the top row?
.(Class 10 Maths Sample Question Paper)
.(Class 10 Maths Sample Question Paper)
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Here, logs are stacked in each row to form a series.21+20+19+28+17+…….
For this A.P.,
First term (a )= 21
d = 20-21 = −1
Let the total number of 200 logs be placed in n rows.
Sn = 216
216= n/2 (2×21+( n -1)(-1)
216 × 2 = n[42- n+1)]
432 = n[42- n+1)]
432 = 42n - n² +n
432 = 43n - n²
n²− 43n + 432= 0
n² -27n -16n +432= 0
n (n−27) −16(n-27)=0
(n − 16) (n − 27) = 0
(n − 16) = 0 or n − 27 = 0
n = 16 or n = 27
an = a + (n − 1)d
For n= 16
a16 = 21 + (16 − 1) (−1)
a16 = 21−15 = 6
a16 = 6
For n = 27
a27 = 21 + (27− 1) (−1)
a27 = 21 − 26
a27= −5
the number of logs in 16th row is 6 & the number of logs in 27th row is negative(numbers of logs in any row can't be negative).
Hence, 216 logs can be placed in 16 rows and the number of logs in the 16th row(top row) is 6.
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