the 4th term of an arithmetic progression is 27 and the sum of the 1st and 4th term is 42. What is the third term

not solved

To find the third term of the arithmetic progression, we can use the given information about the first term, fourth term, and the sum of the first and fourth term.

Let's denote the first term as 'a', and the common difference between the terms as 'd'.

We know that the fourth term is 27, so the equation for the fourth term can be written as:

a + 3d = 27 ---(1) (since the fourth term is a + 3d, as it is 3 terms after the first term)

We also know that the sum of the first and fourth term is 42, so the equation for the sum of the two terms can be written as:

a + (a + 3d) = 42 ---(2) (the sum of the first and fourth term is a + (a + 3d))

Now, we can solve this system of equations to find the values of 'a' and 'd'.

From equation (2), we simplify it as:
2a + 3d = 42 ---(3)

Next, we can solve equation (1) and equation (3) simultaneously.

Subtracting equation (1) from equation (3), we get:
2a + 3d - (a + 3d) = 42 - 27
a = 15

Now, substituting the value of 'a' into equation (1), we get:
15 + 3d = 27
3d = 27 - 15
3d = 12
d = 4

Since we have found the values of 'a' and 'd', we can find the third term by substituting them into equation (1):

a + 2d = 15 + 2(4)
a + 2d = 15 + 8
a + 2d = 23

Therefore, the third term of the arithmetic progression is 23.

well, clearly, the 1st term is 15, and the difference is 9.