You have 798 cans in 19 rows. If the cans are placed so that each row as 3 fewer cans than the other how many cans are in the bottom row?
This is an arithmetic series
You know S19, n=19, d=-3
You don't know "a".
S19= (n/2)[2a + (n-1)d]
798 = 19/2(2a + 18(-3)]
1596 = 38a - 1026
2633 = 38a
a = 69
So depending on whether your bottom row is the
first or the 69th, it would be either 69 or
69 + 18(-3) = 15
check:
69 + 66 + 63 + ... + 15 for 19 terms
= 19(first+last)/2 = (19)(69+15)/2 = 798
To find the number of cans in the bottom row, we can use the information given in the question.
We are told that there are 798 cans in total and they are arranged in 19 rows. Let's assume the number of cans in the bottom row is "x".
According to the information given, each row has 3 fewer cans than the row above it. So, if we consider the first row, which is the row at the top, it will have x + 2 cans (since it will have 3 fewer cans than the row below it, which is the second row).
The second row will have x + 5 cans (since it will have 3 fewer cans than the row below it, which is the third row), and so on.
Using this pattern, we can determine the number of cans in each row:
Row 1: x + 2 cans
Row 2: x + 5 cans
Row 3: x + 8 cans
...
Row 19: x + 56 cans
Now, summing up the cans in all 19 rows should give us the total number of cans, which is 798:
(x + 2) + (x + 5) + (x + 8) + ... + (x + 56) = 798
To simplify the equation, let's remove the parentheses:
19x + (2 + 5 + 8 + ... + 56) = 798
To simplify further, we can calculate the sum of the numbers inside the parentheses:
2 + 5 + 8 + ... + 56 = (19/2) * (2 + 56) = 19 * 29 = 551
Substituting this back into the equation, we get:
19x + 551 = 798
Now, we can solve for x by isolating it on one side of the equation:
19x = 798 - 551
19x = 247
x = 247/19
x ≈ 13
Therefore, there are approximately 13 cans in the bottom row.