The fifth term of an arithmetic sequence is 23 and 12th term is 72. And they say

1. Determine the first three terms of the sequences and the nth term.
2. What is the value of the 10th term.
3. Which term has a value of 268.

To solve these questions, we need to find the common difference (d) and the first term (a) of the arithmetic sequence. Once we have these values, we can use the formulas to find any term in the sequence.

1. Determining the first three terms of the sequence and the nth term:
We know that the fifth term is 23 and the twelfth term is 72.
Let's construct two equations using the formulas for the terms in an arithmetic sequence:

a + 4d = 23 (equation 1)
a + 11d = 72 (equation 2)

To simplify, let's multiply equation 1 by 11 and equation 2 by 4:
11a + 44d = 253 (equation 3)
4a + 44d = 288 (equation 4)

Subtract equation 4 from equation 3 to eliminate d:
11a - 4a = 253 - 288
7a = -35
a = -5

Now, substitute the value of a into either equation 1 or equation 2 to find d:
-5 + 4d = 23
4d = 28
d = 7

Hence, the first term is -5 and the common difference is 7.

To find the second term (n = 2), we can use the formula:
a_n = a + (n-1)d

a_2 = -5 + (2-1) * 7
a_2 = -5 + 7
a_2 = 2

Similarly, we can find the third term (n = 3):
a_3 = -5 + (3-1) * 7
a_3 = -5 + 14
a_3 = 9

Therefore, the first three terms of the sequence are -5, 2, and 9.
The nth term can be found using the formula:
a_n = a + (n-1)d

So, the nth term is a_n = -5 + (n-1) * 7.

2. To find the 10th term:
Substitute n = 10:

a_10 = -5 + (10-1) * 7
a_10 = -5 + 63
a_10 = 58

Therefore, the 10th term of the sequence is 58.

3. To find which term has a value of 268:
We can set up the equation using the formula for the nth term:

a_n = -5 + (n-1) * 7
268 = -5 + (n-1) * 7

Solve for n by isolating the variable:
263 = (n-1) * 7
263 = 7n - 7
7n = 270
n = 38

Therefore, the term with a value of 268 is the 38th term.