In an arithmetic progression, if 17 is the 3rd term, -25 is the 17th term, then -1 is which term?

The $$n^{th}$$ term of an A.P. = $$a + (n - 1) d$$, where 'a' is the first term , 'n' is the number of terms and 'd' is the common difference.

3rd term, $$A_3 = a + (3 - 1) d = 17$$

=> $$a + 2d = 17$$ -----------------(i)

Similarly, 17th term, $$A_{17} = a + 16d = -25$$ ------------------(ii)

Subtracting equation (i) from (ii), we get :

=> $$(16d - 2d) = -25 - 17$$

=> $$d = \frac{-42}{14} = -3$$

Substituting it in equation (i), => $$a - 6 = 17$$

=> $$a = 17 + 6 = 23$$

Let $$n^{th}$$ term = -1

=> $$a + (n - 1) d = -1$$

=> $$23 + (n - 1) (-3) = -1$$

=> $$(n - 1) (-3) = -1 - 23 = -24$$

=> $$(n - 1) = \frac{-24}{-3} = 8$$

=> $$n = 8 + 1 = 9$$