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One answer is:
t{n} = (59n⁵ - 885n⁴ + 5015n³ - 13275n² + 15446n - 4800)/120
Which gives:
t1 = (59 - 885 + 5015 - 13275 + 15446 - 4800)/120 = 1560/120 = 13
t2 = (59×32 - 885×16 + 5015×8 - 13275×4 + 15446×2 - 4800)/120 = 840/120 = 7
t3 = (59×243 - 885×81 + 5015×27 - 13275×9 + 15446×3 - 4800)/120 = 120/120 = 1
t4 = (59×1024 - 885×256 + 5015×64 - 13275×16 + 15446×3 - 4800)/120 = -600/120 = -5
t5 = (59×3125 - 885×625 + 5015×125 - 13275×25 + 15446×4 - 4800)/120 = -1320/120 = -11
t6 = (59×7776 - 885×1296 + 5015×216 - 13275×36 + 15446×5 - 4800)/120 = 5040/120 = 42
However, I expect your teacher is wanting the much simpler:
t{n} = 19 - 6n
which also gives t{1..5} = {13, 7, 1, -5, -11} but gives a different t6 = -17
The above formulae are only valid for n = 1, 2, ..., 5 as t6 is different.
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