Is \(0\) a term of the AP \(31, 28, 25, \ldots\)? Justify your answer.
No.
Step 1: In an Arithmetic Progression (AP), the first term is written as \(a\) and the common difference as \(d\).
Here, the first term \(a = 31\).
The common difference \(d = 28 - 31 = -3\).
Step 2: The formula for the \(n\)-th term of an AP is:
\(a_n = a + (n - 1) \times d\).
Step 3: We want to check if \(0\) is a term in this AP. So, put \(a_n = 0\).
\(0 = 31 + (n - 1)(-3)\)
Step 4: Simplify the equation:
\(0 = 31 - 3(n - 1)\)
\(0 = 31 - 3n + 3\)
\(0 = 34 - 3n\)
Step 5: Solve for \(n\):
\(3n = 34\)
\(n = \dfrac{34}{3}\)
Step 6: Since \(n\) must be a whole number (like 1, 2, 3, …) for the term to exist, and \(\tfrac{34}{3}\) is not a whole number, it means there is no such term.
Therefore, 0 is not a term of the given AP.