The sum of the 2nd and 5th terms of an A. P is 42 if the difference between the 6th and the 3rd terms is 12 find the (a)common difference (b) first term (c) 20th term ?

Let the first term of the A.P be 'a' and the common difference be 'd'.

The 2nd term = a + d
The 5th term = a + 4d
Given that the sum of the 2nd and 5th terms is 42:
(a + d) + (a + 4d) = 42
2a + 5d = 42 ----(1)

The 3rd term = a + 2d
The 6th term = a + 5d
Given that the difference between the 6th and 3rd terms is 12:
(a + 5d) - (a + 2d) = 12
3d = 12
d = 4

Substitute d = 4 in equation (1):
2a + 5(4) = 42
2a + 20 = 42
2a = 22
a = 11

(a) The common difference is 4.
(b) The first term is 11.

The nth term of an A.P is given by:
Tn = a + (n-1)d

Substitute a = 11 and d = 4 in the formula to find the 20th term:
T20 = 11 + (20-1)4
T20 = 11 + 76
T20 = 87

(c) The 20th term of the A.P is 87.