A child organizes animal blocks in rows on the ground. There are 64 animal blocks in the 5th row & 92 of those blocks in the 9th and last row. Suppose that the number of animal blocks frm row to row form an arithmetic sequence.
The number of animal blocks in the first row is....
I can’t seem to find a pattern, having a hard time solving this problem too.
Also How do I find the total number of blocks used in the arrangement?
To find the number of animal blocks in the first row, we can use the formula for an arithmetic sequence.
In an arithmetic sequence, the nth term (aₙ) can be found using the formula:
aₙ = a₁ + (n-1)d
Where:
aₙ is the nth term in the sequence
a₁ is the first term in the sequence
n is the position of the term in the sequence
d is the common difference between terms
In this case, we know the following information:
a₅ = 64 (number of animal blocks in the 5th row)
a₉ = 92 (number of animal blocks in the 9th row)
We want to find a₁, the number of animal blocks in the first row.
Using the formula, we can set up two equations:
64 = a₁ + (5-1)d
92 = a₁ + (9-1)d
Simplifying these equations, we get:
64 = a₁ + 4d
92 = a₁ + 8d
We now have a system of equations. We can solve for a₁ by eliminating the variable d.
Multiplying the first equation by 2 and subtracting the second equation from it, we get:
2(64) - 92 = 2(a₁ + 4d) - (a₁ + 8d)
128 - 92 = 2a₁ + 8d - a₁ - 8d
36 = a₁
Therefore, the number of animal blocks in the first row is 36.
To find the total number of blocks used in the arrangement, we can calculate the sum of the arithmetic sequence.
The sum of an arithmetic sequence can be found using the formula:
Sₙ = (n/2)(a₁ + aₙ)
Where:
Sₙ is the sum of the first n terms
n is the number of terms
a₁ is the first term
aₙ is the nth term
In this case, we want to find the sum of all the animal blocks used in the arrangement, so n would represent the number of rows.
The sum of the n terms can be calculated as follows:
Sₙ = (n/2)(a₁ + aₙ)
S₉ = (9/2)(a₁ + a₉)
Substituting the known values:
S₉ = (9/2)(36 + 92)
S₉ = (9/2)(128)
S₉ = (9 * 128) / 2
S₉ = 576
Therefore, the total number of animal blocks used in the arrangement is 576.
To find the number of animal blocks in the first row, we can use the concept of an arithmetic sequence. In an arithmetic sequence, the difference between each term is constant. Let's denote the first row as the first term (a₁) and the common difference between each row as (d).
Given that the 5th row has 64 animal blocks and the 9th row has 92 animal blocks, we can write two equations based on this information:
a₁ + 4d = 64 (since the 5th row is 5 - 1 = 4 rows away from the first row)
a₁ + 8d = 92 (since the 9th row is 9 - 1 = 8 rows away from the first row)
To solve for a₁, we need to eliminate the variable d. We can do this by subtracting the first equation from the second equation:
(a₁ + 8d) - (a₁ + 4d) = 92 - 64
4d = 28
d = 7
Now that we have found the value of d, we can substitute it back into any of the original equations to solve for a₁. Let's use the first equation:
a₁ + 4d = 64
a₁ + 4(7) = 64
a₁ + 28 = 64
a₁ = 64 - 28
a₁ = 36
Therefore, the number of animal blocks in the first row is 36.
To find the total number of blocks used in the arrangement, we can use the formula for the sum of an arithmetic series:
Sn = (n/2)(a₁ + an)
where Sn represents the sum of the first n terms, a₁ is the first term, and an is the last term.
Given that the 9th and last row has 92 animal blocks, we know that the last term is 92. And since the given information tells us that the arrangement forms an arithmetic sequence, we can determine the total number of rows as well. The number of rows can be calculated by subtracting the row number of the first row (1) from the row number of the last row (9):
Total number of rows = 9 - 1 + 1 = 9
Plugging the values into the formula:
Sn = (n/2)(a₁ + an)
Sn = (9/2)(36 + 92)
Sn = (9/2)(128)
Sn = 9 * 64
Sn = 576
Therefore, the total number of blocks used in the arrangement is 576.