In an arithmetic progression, if 9 is the 5th term, -26 is the 12th term, then -6 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.

5th term, $$A_5 = a + (5 - 1) d = 9$$

=> $$a + 4d = 9$$ -----------------(i)

Similarly, 12th term, $$A_{12} = a + 11d = -26$$ ------------------(ii)

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

=> $$(11d - 4d) = -26 - 9$$

=> $$d = \frac{-35}{7} = -5$$

Substituting it in equation (i), => $$a - 20 = 9$$

=> $$a = 9 + 20 = 29$$

Let $$n^{th}$$ term = -6

=> $$a + (n - 1) d = -6$$

=> $$29 + (n - 1) (-5) = -6$$

=> $$(n - 1) (-5) = -6 - 29 = -35$$

=> $$(n - 1) = \frac{-35}{-5} = 7$$

=> $$n = 7 + 1 = 8$$