Question

Which term of an AP is 21, 42, 63, … is 210?
$\begin{align}
  &{\text{A}}.\;{9^{th}} \\
  &{\text{B}}.\;{10^{th}} \\
  &{\text{C}}.\;{12^{th}} \\
  &{\text{D}}.\;{11^{th}} \\
\end{align} $

Answer 1

Hint: The concept of arithmetic progression will be used in this problem. We will try to find the different parameters of the given AP. The first term is 21 and the common difference is 42 - 21 = 21. Then we will assume a general term for 210 and then substitute the parameters to find the term. The formula for the general term is-
${a_n} = a + \left( {n - 1} \right)d\;$where a is the first term and d is the common difference.

Complete step-by-step answer:
We have been given the AP 21, 42, 63… We will first find the first term a and the common difference d given by the formula-
$d = {a_2} - a$
d = 42 - 21 = 21
Now let us assume that 210 is the ${n^{th}}$ term of the AP. We will use the formula for the general term of the AP, to find the value of n, which will give us the final answer. We can write that-
${a_n} = a + \left( {n - 1} \right)d\;$
We will substitute a = 21, d = 21 and an = 210
$\begin{align}
  &210 = 21 + \left( {n - 1} \right)21 \\
  &189 = 21\left( {n - 1} \right) \\
  &n - 1 = \dfrac{{189}}{{21}} = 9 \\
  &n = 10 \\
\end{align} $

This is the value of n. This means that 210 is the ${10^{th}}$ term of the given AP. The correct option is B. ${10^{th}}$.

Note: The most common mistake in such types of problems is that they may confuse AP with GP. This is because the second term 42 is twice the first term 21. So, this may lead to the confusion that this is a GP with the common ratio 2. Hence, we should always verify the ratio or difference.