Find the number of cans stacked in a display if there are 30 rows of cans with six on the top row, ten on the second row, 14 on the third row and so on.
I know it's the sum of an arithmetic sequence, but I'm not sure exactly what to do.
The highest row has 6 cans
The lowest row (29 rows down from the top) has 6 + (29*4) = 122 cans
The average number in a row is (122+6)/2 = 64 cans.
The total number of cans is 30x64 = 192
To determine the number of cans stacked in the display, we need to find the sum of the arithmetic sequence. An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant.
In this case, the common difference between consecutive terms is 4 because we are adding 4 cans to each row. To find the number of cans in each row, we can use the formula for the nth term of an arithmetic sequence:
n = a + (n - 1)d
where n is the term number, a is the first term, and d is the common difference.
In this sequence, the first term (a) is 6, and the common difference (d) is 4. We need to find the number of cans in the 30th row, so n = 30. Plugging these values into the formula:
30 = 6 + (30 - 1)4
Simplifying:
30 = 6 + 29 * 4
30 = 6 + 116
30 = 122
So, there are 122 cans in the 30th row.
Now, to find the total number of cans stacked in the display, we need to sum the number of cans in each row. We can use the formula for the sum of an arithmetic sequence:
S = (n/2)(2a + (n-1)d)
where S is the sum, n is the number of terms, a is the first term, and d is the common difference.
In this case, we want to find the sum of the first 30 terms, so n = 30. Plugging in the values:
S = (30/2)(2(6) + (30-1)4)
S = 15(12 + 29*4)
S = 15(12 + 116)
S = 15 * 128
S = 1920
Therefore, the total number of cans stacked in the display is 1920.