tn = 1.5*n2 - 1.5*n + 1
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The given sequence is 11, 31, 51, 72 The nth term of this sequence can be expressed as an = 11 + (n - 1) × 20 Therefore, the nth term is 11 + (n - 1) × 20, where n is the position of the term in the sequence.
When n = 1, sequence term (call it y) = 7. Each unit increase in n results in an increase of 6 in y, so when n = 5, y = 31. Sequence is therefore 7, 13, 19, 25, 31 etc.
Assuming this is a linear or arithmetic sequence, the nth term is Un = 31 - 8n. But, there are infinitely many polynomials of order 5 or higher, and many other functions that will fit the above 5 numbers.
There are very many possible solutions. The simplest polynomial solution is t(n) = (n^4 - 14n^3 + 71n^2 + 14n + 24)/24.
There are infinitely many possible answers. the simplest polynomial is Un = -n4 + 12n3 - 47n2 + 78n - 41