the first of a linear sequence is 3 and the 8th term is 31 find the common difference

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3, 3+d, 3+2d , 3+3d ....... 3+7d

3+7d = 31

To find the common difference in a linear sequence, we need to use the formula for the nth term of an arithmetic sequence:

nth term = first term + (n - 1) * common difference

Here, the given information is:
First term (a1) = 3
8th term (a8) = 31

Let's substitute these values into the formula to form two equations:

a1 = 3
a8 = 3 + (8 - 1) * d (where d is the common difference)

By substituting a1 and a8, we can solve for d:

3 + (8 - 1) * d = 31
3 + 7d = 31
7d = 31 - 3
7d = 28
d = 28 / 7
d = 4

Therefore, the common difference in this linear sequence is 4.