Step 1 :The power series is centered at \(x=2\) and it converges for \(x=5\) and diverges for \(x=-5\). This means that the radius of convergence \(R\) is the distance from the center of the series to the point of convergence, which is \(5-2=3\). So, \(R=3\), not \(5\). Therefore, statement P is false.
Step 2 :For statement Q, we need to determine if the series \(\sum_{n=0}^{\infty}(-1)^{n} c_{n}\) converges. This is a series with alternating signs, and we can't determine its convergence without knowing the values of \(c_n\). Therefore, we can't say that statement Q is necessarily true.
Step 3 :For statement R, we need to determine if the series \(\sum_{n=0}^{\infty} 7^{n} c_{n}\) diverges. This is a power series with a common ratio of \(7\), which is greater than \(1\). According to the ratio test, a power series converges if the absolute value of the common ratio is less than \(1\) and diverges if it is greater than \(1\). Therefore, statement R is true.
Step 4 :So, the final answer is R only.
Step 5 :Final Answer: \(\boxed{\text{R only}}\)