Alka loves to organize cans in triangles piles, where each row has one can less than the
row below and there is only one can in the top row. To arrange 10 cans in total, she
places 4 cans in the bottom row and 3 cans in the row above it and so on:
i. Alka now arranges a pile having 20 cans in the bottom row, ending with one can in
the top row. The total number of cans in the arrangement are:__________________
ii. There are 820 cans in a pile. How many cans are there in the bottom now?
________________________
iii. Alka has 200 cans. Explain why she will not be able to organize the cans in a
triangular pile.
________________________
Answers
Given : Alka loves to organize cans in triangles piles, where each row has one can less than the row below and there is only one can in the top row.
Alka arranges a pile having 20 cans in the bottom row, ending with one can in the top row.
To Find : The total number of cans in the arrangement
There are 820 cans in a pile. How many cans are there in the bottom ?
Alka has 200 cans. Explain why she will not be able to organize the cans in a
triangular pile.
Solution:
Bottom row = 20
Next row = 19
last row = 1
This is an AP
with a = 20
L = 1
d = - 1
n = 20
Sum = (n/2)(a + L )
= (20/2)(1 + 20)
= 210
210 Cans
Let say n rows
Then bottom row has n and top row has 1
sum = n(n + 1)/2
n(n + 1)/2 = 820
=> n² + n = 1640
=> n² + 41 n - 40n - 1640 = 0
=> n(n + 41) - 40(n + 41) = 0
=> (n - 40)(n + 41) = 0
=> n = 40
40 cans in bottom row
n(n + 1)/2 = 200
=> n(n + 1) = 400
19(20) = 380
20(21) = 420
so Either 190 or 210 cans be arranged not 200
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