A display of cans on a grocery shelf consists of 20 cans on the bottom, 18 cans in the next row, and so on in an arithmetic sequence, until the top row has 4 cans. How many cans, in total, are in the display?
terms of the sequence:
20 , 18 , 16 ,...4
So we have a = 20, d = -2 , t(n) = 4
t(n) = a + (n-1)d
4 = 20 + (n-1)(-2)
-16 = -2n + 2
2n = 18
n = 9
so we need sum of 9 terms
= (9/2)(first + last)= (9/2)(20 + 4) = 108
or
Sum(n) = (n/2)(2a + (n-1)d)
= (9/2)(40 + 8(-2)) = 108
check:
term9 = a = 8d
= 20 + 8(-2) = 20-16 = 4
One kind of fruit on the fisrt day of January, two kinds on the second, 3 kinds on the third day, and son how many fruit ate until the twelfh?
To find the total number of cans in the display, we need to sum up the cans from all the rows in the arithmetic sequence.
The number of cans in each row follows an arithmetic sequence, starting from 20 and decreasing by 2 each row until reaching 4.
We can use the formula for the sum of an arithmetic sequence to find the total number of cans in the display.
The formula for the sum of an arithmetic sequence is:
Sn = (n/2)(2a + (n-1)d)
Where:
Sn is the sum of the sequence,
n is the number of terms in the sequence,
a is the first term, and
d is the common difference between terms.
Here, the first term a is 20, and the common difference d is -2 (since it decreases by 2 each row).
We need to find the number of terms in the sequence. Let's call it n.
Using the formula for the nth term of an arithmetic sequence:
an = a + (n-1)d
Here, the nth term an is 4 (since the top row has 4 cans).
Plugging in the values:
4 = 20 + (n-1)(-2)
4 = 20 - 2n + 2
-2n + 6 = 0
-2n = -6
n = 3
So, the number of terms in the sequence is 3.
Now, we can plug these values into the formula for the sum of an arithmetic sequence:
Sn = (n/2)(2a + (n-1)d)
Sn = (3/2)(2(20) + (3-1)(-2))
Sn = (3/2)(40 + 2(-2))
Sn = (3/2)(40 - 4)
Sn = (3/2)(36)
Sn = 54
Therefore, there are 54 cans in total in the display.
To find the total number of cans in the display, we need to find the sum of the arithmetic sequence.
We know that the bottom row has 20 cans and the top row has 4 cans. The difference between consecutive terms in the sequence is 18 - 20 = -2.
Let's use the formula for the sum of an arithmetic sequence:
S = (n/2)(a + l)
where S is the sum, n is the number of terms, a is the first term, and l is the last term.
First, let's find the number of terms (n) in the sequence. The formula for the number of terms in an arithmetic sequence is:
n = (l - a)/d + 1
where d is the common difference.
In our case, a = 20, l = 4, and d = -2.
n = (4 - 20)/(-2) + 1
= (-16)/(-2) + 1
= 8 + 1
= 9
So, there are 9 terms in the sequence.
Now, let's find the sum (S) using the formula:
S = (n/2)(a + l)
= (9/2)(20 + 4)
= (9/2)(24)
= 9 * 12
= 108
Therefore, the total number of cans in the display is 108.