一條關於sequence的問題

The general term is 3n^3-2n

If we substitute n=7,

it is 3*7^3-2*7

=343*3-14

=1029-14

=1015//


If we substitute n=8,

it is 3*8^3-2*8

=3*512-2*8

=1536-16

=1520//

1 = 1 × 1
20 = 2 × (1 + 3 × 3)
75 = 3 × (1 + 3 × 3 + 3 × 5)
184 = 4 × (1 + 3 × 3 + 3 × 5 + 3 × 7)
365 = 5 × (1 + 3 × 3 + 3 × 5 + 3 × 7 + 3 × 9)
636 = 6 × (1 + 3 × 3 + 3 × 5 + 3 × 7 + 3 × 9 + 3 × 11)

但這樣的關係確實砌唔到general term，所以要轉一轉：

1 = 1 × (3 × 1 - 2)
20 = 2 × (3 × 1 + 3 × 3 - 2)
75 = 3 × (3 × 1 + 3 × 3 + 3 × 5 - 2)
184 = 4 × (3 × 1 + 3 × 3 + 3 × 5 + 3 × 7 - 2)
365 = 5 × (3 × 1 + 3 × 3 + 3 × 5 + 3 × 7 + 3 × 9 - 2)
636 = 6 × (3 × 1 + 3 × 3 + 3 × 5 + 3 × 7 + 3 × 9 + 3 × 11 - 2)

Let T(n) be the general term
∴The general term
= n × ((3 × 1 + 3 × 3 + 3 × 5 + 3 × 7 + 3 × 9 + 3 × 11 + …… + 3 × (2n - 1)) - 2)
= n × (3 × (1 + 3 + 5 + 7 + 9 + 11 + …… + (2n - 1)) - 2)
= n × (3n² - 2)
= 3n³ - 2n

∴第七個數目
= 3 × 7³ - 2 × 7
= 1015

∴第八個數目
= 3 × 8³ - 2 × 8
= 1520

1) 1 = 1 x 1

2) 20 = 2 x (1 + 3 x 3)

3)75 = 3 x (1 + 3 x 3 + 3 x 5)

4)184 = 4 x ( 1 + 3 x 3 + 3 x 5 + 3 x 7)

5)365 = 5 x ( 1 + 3 x 3 + 3 x 5 + 3 x 7 + 3 x 9)

6)636 = 6 x ( 1 + 3 x 3 + 3 x 5 + 3 x 7 + 3 x 9 + 3 x 11)

7) ? = 7 x ( 1 + 3 x 3 + 3 x 5 + 3 x 7 + 3 x 9 + 3 x 11 + 3 x 13)
= 7 x 145 = 1015

8) ? = 8 x ( 1 + 3 x 3 + 3 x 5 + 3 x 7 + 3 x 9 + 3 x 11 +
3 x 13 + 3 x 15)
= 8 x 190 = 1520