The 4th term of an A.P is 37 and the 6th term is 12 more than the 4th term. Find the seventh term

6th term is 12 more than the 4th term

so, 12 = 2d, and d=6

a_7 = a_4 + 3d = 37 + 3*6 = 55

To find the seventh term of an arithmetic progression (A.P.), we need to first determine the common difference of the A.P.

Given that the fourth term is 37, we can denote it as a₄ = 37.

Next, we are given that the sixth term is 12 more than the fourth term. We can represent this as a₆ = a₄ + 12.

We know that the formula to find the nth term of an arithmetic progression is given by:
aₙ = a + (n - 1)d
where a is the first term, n is the term number, and d is the common difference.

In this case, we need to find the common difference (d) so that we can find the seventh term.

Using the information from the given conditions, we can set up two equations:

a₄ = a + 3d (since the fourth term is a + 3d)
a₆ = a + 5d (since the sixth term is a + 5d)

From equation (1): a₄ = a + 3d
Substitute a₄ = 37: 37 = a + 3d

From equation (2): a₆ = a + 5d
Substitute a₆ = a₄ + 12: a₄ + 12 = a + 5d
Substitute a₄ = 37: 37 + 12 = a + 5d
49 = a + 5d

Now we have two equations:
37 = a + 3d (Equation 1)
and
49 = a + 5d (Equation 2)

To solve these simultaneous equations, we can subtract Equation 1 from Equation 2:

49 - 37 = (a + 5d) - (a + 3d)
12 = 2d
d = 6

Now that we have found the common difference, d = 6, we can substitute it back into one of the equations, such as Equation 1, to find the value of a:

37 = a + 3(6)
37 = a + 18
a = 37 - 18
a = 19

So the first term (a) is 19, and the common difference (d) is 6.

Now we can find the seventh term (a₇) using the formula:
aₙ = a + (n - 1)d

Substituting a = 19, n = 7, and d = 6 into the formula:
a₇ = 19 + (7 - 1) * 6
a₇ = 19 + 6 * 6
a₇ = 19 + 36
a₇ = 55

Therefore, the seventh term of the arithmetic progression is 55.

I have no idea

Pls help me solve it.