Step 1 :Given a geometric sequence with \(a_{1}=5\) and \(a_{4}=-5\).
Step 2 :A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the ratio.
Step 3 :We can use the formula for the nth term of a geometric sequence, which is \(a_{n}=a_{1} \times r^{(n-1)}\), where \(r\) is the common ratio.
Step 4 :We can find the common ratio by substituting \(n=4\) and \(a_{4}=-5\) into the formula and solving for \(r\).
Step 5 :Substituting the values, we get \( -5 = 5 \times r^{(4-1)} \). Solving this equation, we find that \(r = -1\).
Step 6 :Once we have the common ratio, we can find \(a_{7}\) by substituting \(n=7\) and the found \(r\) into the formula.
Step 7 :Substituting the values, we get \(a_{7} = 5 \times (-1)^{(7-1)}\).
Step 8 :Solving this equation, we find that \(a_{7} = 5\).
Step 9 :Final Answer: The seventh term of the geometric sequence is \(\boxed{5}\).