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 need to determine the common difference between each row. Since the problem states that the number of blocks from row to row forms an arithmetic sequence, we can use this information to solve for the first row.
Let's denote the number of blocks in the first row as 'a', and the common difference as 'd'. From the given information:
Number of blocks in the 5th row = 64
Number of blocks in the 9th row = 92
The formula for the nth term of an arithmetic sequence is given by:
an = a + (n - 1) * d
Using this formula, we can substitute the values for the 5th and 9th rows:
64 = a + (5 - 1) * d
92 = a + (9 - 1) * d
Simplifying these equations, we get:
64 = a + 4d -- Equation 1
92 = a + 8d -- Equation 2
Now we have a system of equations with two variables (a and d). We can solve this system of equations to find the values of a and d.
Subtracting Equation 1 from Equation 2, we eliminate 'a':
92 - 64 = (a + 8d) - (a + 4d)
28 = 4d
d = 7
Substituting the value of d back into Equation 1, we can find 'a':
64 = a + 4 * 7
64 = a + 28
a = 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 need to sum the number of blocks in each row. Since we know the number of blocks in the first row (36) and the common difference (7), we can find the number of blocks in each row using the formula:
Number of blocks in nth row = a + (n - 1) * d
Where 'a' is the number of blocks in the first row and 'd' is the common difference.
To find the total number of blocks, we can sum up the number of blocks in each row from the first to the last row. In this case, the first row is the 1st row and the last row is the 9th row.
Total number of blocks = (a + (1 - 1) * d) + (a + (2 - 1) * d) + ... + (a + (9 - 1) * d)
Now we can substitute the values of 'a' and 'd' into the formula and calculate the sum to find the total number of blocks.
To find the number of animal blocks in the first row, you can use the information given about the arithmetic sequence.
Let's assume that the common difference between each row is denoted by 'd'. The formula for the nth term of an arithmetic sequence is: an = a1 + (n-1) * d, where 'a1' is the first term of the sequence and 'n' is the number of terms.
In this case, the 5th row has 64 blocks and the 9th row has 92 blocks. Using the formula, we can write two equations:
a5 = a1 + (5-1) * d = 64
a9 = a1 + (9-1) * d = 92
Simplifying these equations, we get:
a1 + 4d = 64 --> Equation 1
a1 + 8d = 92 --> Equation 2
You now have a system of two equations with two unknowns (a1 and d). By solving this system, you can find the values of a1 and d.
Subtracting Equation 1 from Equation 2, we get:
a1 + 8d - (a1 + 4d) = 92 - 64
4d = 28
Dividing both sides by 4, we find the value of d:
d = 28 / 4
d = 7
Now that we have the value of d, we can substitute it into either Equation 1 or 2 to find a1. Let's use Equation 1:
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, you can use the formula for the sum of an arithmetic series. The formula is: Sn = (n/2) * (2a1 + (n-1) * d), where 'Sn' is the sum of the series, and 'n' is the number of terms.
In this case, the last row has 9 terms, and the first row has 1 term. Substitute these values into the formula:
S9 = (9/2) * (2 * 36 + (9 - 1) * 7)
S9 = (9/2) * (72 + 8 * 7)
S9 = (9/2) * (72 + 56)
S9 = (9/2) * 128
S9 = 9 * 64
S9 = 576
Therefore, the total number of blocks used in the arrangement is 576.