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 ? ? ?
Is the pattern 6^3 (4x7^2)+(3x6)+2 216
7^3 (5x8^2)+ (3x7)+2 343
8^3 (6x9^2)+ (3x8)+2 512
9^3 (7X10^2)+(3x9)+2 729
10^3(8x11^2)+(3x10)+2 1000
11^3 (9x12^2)+(3x11)+2 1331
n^3 (n-2)x( )^2)+(3x )+2 n^3 how do i come up with the pattern for this?
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To come up with the pattern for the given table, let's analyze the calculations for each row:
First Row:
2^3 = 8
(0 x 3^2) + (3 x 2) + 2 = 2 + 6 + 2 = 10
Second Row:
3^3 = 27
(1 x 4^2) + (3 x 3) + 2 = 16 + 9 + 2 = 27
Third Row:
4^3 = 64
(2 x 5^2) + (3 x 4) + 2 = 50 + 12 + 2 = 64
Fourth Row:
5^3 = 125
(3 x 6^2) + (3 x 5) + 2 = 108 + 15 + 2 = 125
Based on these calculations, we can observe the following pattern:
For each row, the first element is equal to n^3 (where n is the row number).
The second element consists of two terms: (n-2) multiplied by the square of a number ( ), and 3 multiplied by the row number.
The third element is simply 2.
Now let's find the missing elements in the pattern:
Sixth Row:
6^3 = 216
(4 x 7^2) + (3 x 6) + 2 = 4 x 49 + 18 + 2 = 216
Seventh Row:
7^3 = 343
(5 x 8^2) + (3 x 7) + 2 = 5 x 64 + 21 + 2 = 343
Eighth Row:
8^3 = 512
(6 x 9^2) + (3 x 8) + 2 = 6 x 81 + 24 + 2 = 512
Ninth Row:
9^3 = 729
(7 x 10^2) + (3 x 9) + 2 = 7 x 100 + 27 + 2 = 729
Tenth Row:
10^3 = 1000
(8 x 11^2) + (3 x 10) + 2 = 8 x 121 + 30 + 2 = 1000
Eleventh Row:
11^3 = 1331
(9 x 12^2) + (3 x 11) + 2 = 9 x 144 + 33 + 2 = 1331
Therefore, the pattern for the entire table is:
n^3, (n-2) multiplied by the square of a number, 3 multiplied by the row number, and 2.