Final answer:
To find the common difference, we utilized the properties of an arithmetic sequence and the given conditions. By expressing the terms a_7 and a_4 in terms of the first term and the common difference, and substituting a_3's value, we deduced that the common difference is 1/2.
Step-by-step explanation:
To find the common difference in an arithmetic sequence given a_7 - 2a_4 = 1 and a_3 = 0, we need to use the definition of an arithmetic sequence, which has the form: a_n = a_1 + (n-1)d, where a_n is the nth term, a_1 is the first term, d is the common difference, and n is the term number.
First, we express a_7 and a_4 in terms of a_1 and d:
- a_7 = a_1 + 6d
- a_4 = a_1 + 3d
Substituting these into the given equation:
(a_1 + 6d) - 2(a_1 + 3d) = 1
Expanding and simplifying gives:
a_1 + 6d - 2a_1 - 6d = 1
-a_1 = 1
We also know that a_3 = 0, so:
a_3 = a_1 + 2d = 0
Substituting a_1 = -1 into the expression for a_3, we get:
-1 + 2d = 0
Therefore, the common difference d is:
d = 1/2