The 9th term of an arithmetic sequence is 12 and the 17th term is 28, find the 4th term of the arithmetic sequences

9th term = a+8d = 12

17th term= a+16d = 28
subtract them
8d = 16
d = 2

sub into a+8d = 12
a = -4

4th term = a+3d = -4 + 6 = 2

To find the 4th term of the arithmetic sequence, we can use the formula for the nth term of an arithmetic sequence:

a_n = a_1 + (n - 1)d
where:
a_n is the nth term of the sequence,
a_1 is the first term of the sequence, and
d is the common difference.

We are given that the 9th term of the sequence is 12, so we can plug this into the formula and solve for a_1 + 8d = 12.

Similarly, the 17th term of the sequence is 28, so we can plug this into the formula and solve for a_1 + 16d = 28.

Now we have a system of two equations with two unknowns (a_1 and d):

a_1 + 8d = 12 ----(1)
a_1 +16d = 28 ----(2)

To solve this system, we can subtract equation (1) from equation (2) to eliminate a_1:

(a_1 + 16d) - (a_1 + 8d) = 28 - 12

Simplifying this, we get:

8d = 16

Dividing both sides by 8, we find:

d = 2

Now, we can substitute this value of d into equation (1) to solve for a_1:

a_1 + 8(2) = 12
a_1 + 16 = 12
a_1 = -4

Therefore, the first term of the arithmetic sequence (a_1) is -4 and the common difference (d) is 2.

Finally, we can find the 4th term of the sequence using the formula:

a_4 = -4 + (4 - 1)2

Simplifying this, we get:

a_4 = -4 + 3(2)
a_4 = -4 + 6
a_4 = 2

Therefore, the 4th term of the arithmetic sequence is 2.

To find the 4th term of the arithmetic sequence, we need to determine the common difference (d) of the sequence.

Given that the 9th term of the sequence is 12, we can use this information to find the value of the 1st term (a).

Using the formula for arithmetic sequences, which states that the nth term (Tn) can be calculated using the formula: Tn = a + (n - 1)d, we can substitute n = 9 and Tn = 12:

12 = a + (9 - 1) * d
12 = a + 8d

Now, let's use the second piece of information: the 17th term is 28. Again using the formula, this time substituting n = 17 and Tn = 28:

28 = a + (17 - 1) * d
28 = a + 16d

We have two equations:

a + 8d = 12 ---> (Equation 1)
a + 16d = 28 ---> (Equation 2)

We can solve this system of equations to find the values of a and d. Subtracting Equation 1 from Equation 2, we have:

(a + 16d) - (a + 8d) = 28 - 12
8d = 16
d = 2

Now that we have the value of the common difference (d = 2), we can substitute it back into Equation 1 to find the value of a:

a + 8(2) = 12
a + 16 = 12
a = -4

Now we can use the formula for the nth term to find the 4th term by substituting n = 4, a = -4, and d = 2:

T4 = a + (4 - 1) * d
T4 = -4 + 3 * 2
T4 = -4 + 6
T4 = 2

Therefore, the 4th term of the arithmetic sequence is 2.