any help please?
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Make a table.
.. n ..... a(n)
1 ......... -2
2 .......... 6
3 ........ -12
4 ......... 20
5 ........ -30
Notice that there are 2 patterns here- One is alternating negative signs, which suggests (-1)^n.
The other pattern is the set of differences 4, 6, 8, 10...
The second level of differences is a constant 2. This means the formula has 2n²/2 = n² in it.
Add another column to the table for n², and see what is necessary to get to a(n).
.. n ..... a(n) .... n² ... ?
1 ......... -2 ....... 1 ... +1
2 .......... 6 ....... 4 ... +2
3 ........ -12 ...... 9 ... +3
4 ......... 20
5 ........ -30
n ...... (-1)^n(n² + n) = (-1)^n(n(n + 1))
.. n ..... a(n)
1 ......... -2
2 .......... 6
3 ........ -12
4 ......... 20
5 ........ -30
Notice that there are 2 patterns here- One is alternating negative signs, which suggests (-1)^n.
The other pattern is the set of differences 4, 6, 8, 10...
The second level of differences is a constant 2. This means the formula has 2n²/2 = n² in it.
Add another column to the table for n², and see what is necessary to get to a(n).
.. n ..... a(n) .... n² ... ?
1 ......... -2 ....... 1 ... +1
2 .......... 6 ....... 4 ... +2
3 ........ -12 ...... 9 ... +3
4 ......... 20
5 ........ -30
n ...... (-1)^n(n² + n) = (-1)^n(n(n + 1))