Given the sequence defined by $a_n=\cos \left(n\pi \right)$, which of the following lists the first five terms of the sequence?

• A. $1,0,-1,0,1$
• B. $1,1,1,1,1$
• C. $0,1,0,-1,0$
• D. $1,-1,1,-1,1$

Answer: (D)

Verified step by step guidance

1. Identify the general term of the sequence: \(a_n = \cos(n\pi)\), where \(n\) is a positive integer starting from 1.
2. Recall the key trigonometric identity: \(\cos(\pi) = -1\), and more generally, \(\cos(k\pi) = (-1)^k\) for any integer \(k\).
3. Apply this identity to the sequence term: \(a_n = \cos(n\pi) = (-1)^n\).
4. Calculate the first five terms by substituting \(n = 1, 2, 3, 4, 5\) into \(a_n = (-1)^n\):
5. This will give you the sequence: \(a_1 = (-1)^1\), \(a_2 = (-1)^2\), \(a_3 = (-1)^3\), \(a_4 = (-1)^4\), \(a_5 = (-1)^5\), which correspond to alternating values of 1 and -1.