Question

Which term of the following A.P. is 210?
21, 42, 63, 84…….

Answer 1

Hint: To find the order of the term 210 in the above A.P. we have to find the general term of the given A.P. For that we require the first term and the common difference so the first term is the first term of the given A.P. and the common difference is calculated by subtracting any term of the given A.P. from the successor of the selected term. Now, we know the general term of an A.P. is equal to ${{T}_{n}}=a+\left( n-1 \right)d$. In this formula, “a” represents the first term and “d” represents the common difference which we have already calculated above so substitute in this formula. After that, we have to find the order of the term 210 so equate the formula to 210 and then find the value of “n”.

Complete step-by-step solution:
We have given the following A.P.:
21, 42, 63, 84…….
In the above A.P., the first term which we denote by “a” is equal to 21. And the common difference which we denote by “d” is calculated by taking any term and subtracting that term from the term which is the successor to it.
Let us pick a term from the given A.P. as 42 so the successor term from 42 is 63 so subtracting 42 from 63 we get,
$\begin{align}
  & 63-42 \\
 & =21 \\
\end{align}$
Hence, we got the common difference as 21.
Now, we know that the general term of an A.P. is equal to:
${{T}_{n}}=a+\left( n-1 \right)d$
In the above formula, “a” is the first term and “d” is the common difference. As we have calculated the first term and common difference of the given A.P. as 21 and 21 respectively so substituting these values in the above formula we get,
$\begin{align}
  & {{T}_{n}}=21+\left( n-1 \right)21 \\
 & \Rightarrow {{T}_{n}}=21+21n-21 \\
 & \Rightarrow {{T}_{n}}=21n \\
\end{align}$
Now, we have to find the ${{n}^{th}}$ value of the term 210 so equate the above equation to 210 and then solve the equation to get the value of “n”.
$21n=210$
Dividing 21 on both the sides we get,
$n=\dfrac{210}{21}=10$
Hence, ${{10}^{th}}$ term of the above A.P. is equal to 210.

Note: The mistake that could happen in this problem is in writing the formula for ${{n}^{th}}$ term of an A.P.
The formula for writing the general term of an A.P. is equal to:
${{T}_{n}}=a+\left( n-1 \right)d$
Now, the mistake in writing the above formula is that instead of n – 1, you might write n because it will be in sync with the subscript “n” written with “T” so make sure you won’t make this mistake. The point to be pondered upon is that the coefficient of “d” in this formula is “n – 1” not “n”.