the 8th term of an a.p is 37 and 12th term is 57 .find the a.p

please help this questions .please help.

8th term is 37 ----> a+7d = 37

12th term is 57 ---> a + 11d = 57

subtract them:
4d = 20
d = 5

into a+7d=37
a+35=37
a = 2

A.P. is 2 7 12 17 ....

To find the arithmetic progression (AP), we can use the formula for the nth term of an AP:

an = a + (n - 1) * d

where:
an = the nth term,
a = the first term,
n = the position of the term,
d = the common difference.

Using this formula, we can set up two equations:

a8 = a + (8 - 1) * d = 37 ----(1)
a12 = a + (12 - 1) * d = 57 ----(2)

We have a system of two equations with two unknowns (a and d). Let's solve this system:

From equation (1), we have:
a + 7d = 37

From equation (2), we have:
a + 11d = 57

We can solve this system by subtracting equation (1) from equation (2):

(a + 11d) - (a + 7d) = 57 - 37
4d = 20

Divide both sides by 4:
d = 5

Now substitute the value of d in equation (1) to find the value of a:

a + 7d = 37
a + 7 * 5 = 37
a + 35 = 37
a = 37 - 35
a = 2

So, the first term (a) is 2 and the common difference (d) is 5.

Therefore, the arithmetic progression is: 2, 7, 12, 17, 22, 27, 32, 37, 42, 47, 52, 57.

Sure! To find the arithmetic progression (a.p.), we need to determine the common difference and the first term.

First, let's use the formula for the nth term of an arithmetic progression:
nth term = a + (n - 1)d, where "a" is the first term and "d" is the common difference.

We are given that the 8th term is 37, so we can substitute these values into the formula:
37 = a + (8 - 1)d

Simplifying this equation, we have:
37 = a + 7d (Equation 1)

Next, we are given that the 12th term is 57. Substituting these values into the formula:
57 = a + (12 - 1)d

Simplifying, we have:
57 = a + 11d (Equation 2)

Now, we have two equations with two variables (a and d). We can solve these equations simultaneously to find the values of a and d.

Subtracting Equation 1 from Equation 2, we have:
20 = 4d

Dividing both sides of the equation by 4, we get:
d = 5

Now, substitute the value of d back into Equation 1:
37 = a + 7(5)

Simplifying, we have:
37 = a + 35

Subtracting 35 from both sides, we get:
a = 2

Therefore, the first term (a) is 2, and the common difference (d) is 5. Thus, the arithmetic progression is 2, 7, 12, 17, 22, 27, 32, 37, 42, 47, 52, 57.