The sum of 4th and 8th terms of an A.P. is 24 and the sum of the 6th and 10th terms is 34. The First term is:

To solve this problem, we need to use the formula for the nth term of an arithmetic progression (A.P.), which is a + (n-1)d, where a is the first term and d is the common difference.

Given that the sum of the 4th and 8th terms is 24, we can write this as:

a + 3d + a + 7d = 24 2a + 10d = 24 ----(1)

We are also given that the sum of the 6th and 10th terms is 34, which can be written as:

a + 5d + a + 9d = 34 2a + 14d = 34 ----(2)

Now we have a system of two equations, and we can solve it by subtracting equation (1) from equation (2):

2a + 14d - 2a - 10d = 34 - 24 4d = 10 d = 10 / 4 d = 2.5

Substitute d = 2.5 into equation (1):

2a + 10*2.5 = 24 2a + 25 = 24 2a = 24 - 25 2a = -1 a = -1 / 2 a = -0.5

So, the first term of the A.P. is -0.5.