Which term of the AP \(21,42,63,84,\ldots\) is \(210\)?
9th
10th
11th
12th
Step 1: In an Arithmetic Progression (AP), the first term is called a and the common difference is called d.
Here, the AP is \(21, 42, 63, 84, \ldots\).
So, \(a = 21\) and \(d = 42 - 21 = 21\).
Step 2: The formula for the \(n^{\text{th}}\) term of an AP is:
\(a_n = a + (n-1)\,d\)
Step 3: We are asked: which term is \(210\)?
This means we set \(a_n = 210\).
So, \(210 = 21 + (n-1)\times 21\).
Step 4: Simplify the equation:
\(210 = 21 + 21(n-1)\)
\(210 = 21 + 21n - 21\)
\(210 = 21n\)
Step 5: Divide both sides by 21:
\(n = \dfrac{210}{21} = 10\).
Final Answer: The \(210\) is the 10th term of the AP.