If (a_n=5-4n), what is the common difference of this sequence?

Correct answer: (-4)

The direct answer is option A: \(-4\). The rule is \(a_n=5-4n\). In a linear formula of the form \(a_n=c+dn\), the coefficient of \(n\) is the common difference \(d\). We can also prove this by finding consecutive terms. The first term is \(a_1=5-4(1)=1\), the second is \(a_2=5-4(2)=-3\), and the difference is \(-3-1=-4\). The third term is \(-7\), and again \(-7-(-3)=-4\). Thus every step decreases by 4. Option A, \(-4\), is correct because it includes the negative sign. Option B, 4, has the wrong sign: it would describe a sequence increasing by 4. Option C, 5, is only the constant part of the expression; it is not the change from one term to the next. Option D, \(-5\), confuses the constant 5 with the coefficient of \(n\). A negative common difference means the terms go downward, not that the sequence is invalid. For a quick exam method, compare \(a_{n+1}\) and \(a_n\): \(a_{n+1}-a_n=[5-4(n+1)]-[5-4n]=-4\).