a grocery store has a display of cans stacked in layers. The top layer has 1 can. Every other layer has 4 more cans than the layer above it. If there are 12 layers, use the formula for the sum of a finite series to find the number of cans in the display. showing your work is appreciated.
a = 1
d = 4
Sn = n/2 (2+4(n-1))
To find the number of cans in the display, we need to determine the number of cans in each layer and then sum them up using the formula for the sum of a finite series.
Let's start by finding the number of cans in each layer. We are given that the top layer has 1 can.
For the other layers (excluding the top layer), each layer has 4 more cans than the layer above it. We can represent this pattern with the sequence:
1, 5, 9, 13, ...
We can observe that this is an arithmetic sequence with a common difference of 4. We can use the formula for the nth term of an arithmetic sequence to find the number of cans in each layer.
The formula for the nth term of an arithmetic sequence is:
an = a1 + (n - 1)d
Where:
an = nth term of the sequence
a1 = first term of the sequence
n = position of the term in the sequence
d = common difference of the sequence
In our case, a1 = 1 and d = 4.
Using this formula, we can find the number of cans for each layer:
a2 = 1 + (2 - 1) x 4 = 1 + 4 = 5
a3 = 1 + (3 - 1) x 4 = 1 + 8 = 9
a4 = 1 + (4 - 1) x 4 = 1 + 12 = 13
...
We can see that the number of cans in each layer follows the sequence: 1, 5, 9, 13, ...
To find the number of cans in the entire display, we need to sum up the terms of the sequence. We can use the formula for the sum of a finite series to do this.
The formula for the sum of a finite arithmetic series is:
Sn = (n/2)(a1 + an)
Where:
Sn = sum of the series
n = number of terms in the series
a1 = first term of the series
an = last term of the series
In our case, the number of terms in the series (layers) is 12, and we know the first term a1 = 1 and the last term an is the value we need to find.
Using the formula, we can find the sum of the series:
Sn = (12/2)(1 + an)
We know that an = a1 + (n - 1)d, where n = 12 and d = 4.
So, we have:
an = 1 + (12 - 1) x 4 = 1 + 44 = 45
Substituting this value into the formula for the sum of the series:
Sn = (12/2)(1 + 45)
= (6)(46)
= 276
Therefore, the number of cans in the display is 276.
To find the number of cans in the display, we need to determine the number of cans in each layer and then sum them up.
First, let's determine the number of cans in each layer. We are given that the top layer has 1 can. Every other layer has 4 more cans than the layer above it.
Let's use a pattern to find the number of cans in each layer:
Layer 1: 1 can
Layer 2: 1 + 4 = 5 cans
Layer 3: 5 + 4 = 9 cans
Layer 4: 9 + 4 = 13 cans
...
We can see that the number of cans in each layer forms an arithmetic sequence, where each term is 4 more than the previous term.
Now, let's find the number of cans in the 12 layers using the formula for the sum of a finite arithmetic series:
Sum = (n/2) * (first term + last term)
Where:
- n is the number of terms (in this case, the number of layers which is 12)
- first term is the number of cans in the top layer (1 can)
- last term is the number of cans in the 12th layer
To find the last term, which is the number of cans in the 12th layer, we can use the formula for the nth term of an arithmetic sequence:
nth term = a + (n - 1) * d
Where:
- a is the first term (1 can)
- n is the term number (12th layer)
- d is the common difference between terms (4 cans)
Plugging these values into the formula, we get:
nth term = 1 + (12 - 1) * 4
= 1 + 11 * 4
= 1 + 44
= 45 cans
Now, let's substitute the values into the sum formula to find the total number of cans:
Sum = (n/2) * (first term + last term)
= (12/2) * (1 + 45)
= 6 * 46
= 276 cans
Therefore, the number of cans in the display is 276.