Study the number pattern in the table below and complete
2^3 (0x3^2)+(3x2)+2 8
3^3 (1x4^2)+(3x3)+2 27
4^3 (2x5^2)+ (3x4)+2 64
5^3(3x6^2)+(3x5)+2 125
6^3 ? ? ? ?
? ? ? ? ?
10^3 ? ? ?
n^3 (n-2)x( )^2)+(3x )+2 n^3
And what part in this pattern do you not see?
I will complete the n^3 ... row
n^3 (n-2)(n+1)^2 + 3n + 2 n^3
To complete the table and find the missing values, we will examine the number pattern and identify the relationships between the given numbers. Let's break down the given pattern step-by-step:
Step 1: Exponentiation
In the first column, we have n raised to the power of 3 (n^3). The values in this column are obtained by cubing the corresponding n values. For example, 2^3 = 8, 3^3 = 27, 4^3 = 64, and so on.
Step 2: Multiplication and Squaring
In the second column, we have (n-2) multiplied by the square of a certain number, with n as the current row's value. The square of the corresponding (n-2) value is multiplied by this number. For example, for the first row, n-2 = 0, so we multiply 0 by a squared value. In the second row, n-2 = 1, so we multiply 1 by a squared value, and so on.
Step 3: Addition and Multiplication
In the third column, we have the sum of three terms: (3x ), (3x n), and 2. The value of (3x ) is a constant, whereas (3x n) is the product of 3 multiplied by the current row's value (n). We then add these two terms and add 2 to obtain the value in this column.
Using these patterns, we can determine the missing values in the table.
For the missing values in the 6th row, we can follow the same pattern described above:
6^3 = 216
(6-2) = 4
(4 x 7^2) + (3 x 6) + 2 = 196 + 18 + 2 = 216
Therefore, the missing values in the 6th row are: 6^3, 4, 7.
For the missing values in the 7th row, we can follow the same pattern:
7^3 = 343
(7-2) = 5
(5 x 8^2) + (3 x 7) + 2 = 320 + 21 + 2 = 343
Therefore, the missing values in the 7th row are: 7^3, 5, 8.
Similarly, for the missing values in the 8th row:
8^3 = 512
(8-2) = 6
(6 x 9^2) + (3 x 8) + 2 = 486 + 24 + 2 = 512
Therefore, the missing values in the 8th row are: 8^3, 6, 9.
To generalize the pattern for the nth row, the formula would be:
n^3, (n-2), (n+1)
Using this formula, we can calculate the missing values for any row in the table.
Please note that this explanation assumes the given pattern is consistent and the same pattern continues beyond the given rows.