Answers:
1a) The next four terms are: 513, -728, 1001, -1330
1b) The direct formula is [tex]a_n = (-1)^n*n^3+1[/tex]
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Explanation:
It helps to start with part B first. The direct formula will help us find the next four terms in a very efficient manner.
Start with the sequence {0, 9, -26, 65, -124, 217, -342}
Subtract 1 from each term to get this new sequence {-1, 8, -27, 64, -125, 216, -343}, which closely resembles the sequence {1, 8, 27, 64, 125, 216, 343}. This is the sequence of perfect cubes. The only difference is that each term alternates from positive to negative and vice versa.
So we will have an n^3 as part of the equation and also a (-1)^n as part of the equation. The (-1)^n portion allows us to alternate in signs. Put together we have (-1)^n*n^3 so far
The last thing we do is add 1 to this so that we undo the operation "subtract 1" we did earlier to the original list.
Therefore the formula is [tex]a_n = (-1)^n*n^3+1[/tex]
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To help verify we have the right formula, plug in n = 1 and we get
[tex]a_n = (-1)^n*n^3+1[/tex]
[tex]a_1 = (-1)^1*1^3+1[/tex]
[tex]a_1 = 0[/tex]
and plug in n = 2 to get
[tex]a_n = (-1)^n*n^3+1[/tex]
[tex]a_2 = (-1)^2*2^3+1[/tex]
[tex]a_2 = 9[/tex]
and so on. I'll let you check the other terms
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Let's find the terms a8,a9,a10,a11
This is simply a matter of plugging n = 8, n = 9, n = 10, and n = 11
Plug in n = 8
[tex]a_n = (-1)^n*n^3+1[/tex]
[tex]a_8 = (-1)^8*8^3+1[/tex]
[tex]a_8 = 513[/tex]
Repeat for n = 9
[tex]a_n = (-1)^n*n^3+1[/tex]
[tex]a_{9} = (-1)^9*9^3+1[/tex]
[tex]a_{9} = -728[/tex]
Repeat for n = 10
[tex]a_n = (-1)^n*n^3+1[/tex]
[tex]a_{10} = (-1)^{10}*10^3+1[/tex]
[tex]a_{10} = 1001[/tex]
Repeat for n = 11
[tex]a_n = (-1)^n*n^3+1[/tex]
[tex]a_{11} = (-1)^{11}*11^3+1[/tex]
[tex]a_{11} = -1330[/tex]
