# The nth term of an AP is always a linear explanation

This was a good question from Arithmetic Progression which I thought that not much brooding on how to solve it was necessary at all.

The question is: Justify if it is true to say that the following are nth terms of an AP. (i) 2n – 3 (ii) 3n^2 + 5 (iii) 1+n+n^2

I found that that a linear explanation of nth term, like pn+q where p and q are rational numbers can always be put in the form: (q+p) + (n-1)p; which is nth term of an AP with initial term being (q+p) and common difference p.

On the other hand, a quadratic explanation of nth term, like pn^2+rn + q where p, r and q are rational numbers can never be put in the form: a + (n-1)d, as there shall be an exponential increment in every consecutive term due to n^2 is NEVER an AP.

Hence, for given question, (i) is an AP, while (ii) and (iii) are not AP.

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