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.