Final answer:
The recurrence relation for the number of bit sequences with an even number of 0s is A(n) = 2 * A(n-1) with A(1) = 2.
For sequences of length seven, there are 128 such bit sequences.
Step-by-step explanation:
To find a recurrence relation for the number of bit sequences of length n with an even number of 0s, consider two cases: adding a 0 or adding a 1 to a sequence of length n-1.
If the sequence of length n-1 already has an even number of 0s, adding a 1 keeps the number of 0s even. If the sequence has an odd number of 0s, adding a 0 makes it even.
Thus, for any sequence of length n-1 (odd or even), there is exactly one way to extend it to a sequence of length n with an even number of 0s.
Let A(n) denote the number of bit sequences of length n with an even number of 0s.
The recurrence relation can be written as A(n) = 2 * A(n-1), with the base case A(1) = 2, for initial sequences 1 (one 0) and 00 (even number of 0s).
Regarding sequences of length seven, we use the recurrence relation starting from A(1) and find A(7).
Calculating A(7)
A(1) = 2
A(2) = 2 * A(1) = 4
A(3) = 2 * A(2) = 8
...
A(7) = 2 * A(6) = 2 * 64
= 128
Therefore, there are 128 bit sequences of length seven that contain an even number of 0s.