In an arithmetic progression if 13 is the 3rd term, ­47 is the 13th term, then ­30 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, $$A_3 = a + (3 - 1) d = 13$$

=> $$a + 2d = 13$$ -----------------(i)

Similarly, 13th term, $$A_{13} = a + 12d = 47$$ ------------------(ii)

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

=> $$(12d - 2d) = 47 - 13 = 34$$

=> $$d = \frac{34}{10} = 3.4$$

Substituting it in equation (i), => $$a + 2 \times 3.4 = 13$$

=> $$a = 13 - 6.8 = 6.2$$

Let $$n^{th}$$ term = 30

=> $$a + (n - 1) d = 30$$

=> $$6.2 + (n - 1) (3.4) = 30$$

=> $$(n - 1) (3.4) = 30 - 6.2 = 23.8$$

=> $$(n - 1) = \frac{23.8}{3.4} = 7$$

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