Question

What is the sum of the first 14 terms of the AP 12, 15, 18, 21 ……?
A. 144
B. 114
C. 441
D. 414

Answer 1

Hint: For the above question, we will use the formula of the sum of n terms of an AP having the first term as ‘a’ and a common difference ‘d’ which is given by, ${{S}_{n}}=\dfrac{n}{2}\left( 2a+\left( n-1 \right)d \right)$. So, by using this formula, we will get the required sum of the given number of terms of the AP.

Complete step-by-step answer:
In this question, we have been asked to find the sum of the first 14 terms of the AP 12, 15, 18, 21 ……. We know that the sum of n terms of an AP having the first term ‘a’ and a common difference ‘d’ which is given by, ${{S}_{n}}=\dfrac{n}{2}\left( 2a+\left( n-1 \right)d \right)$. So, here, we get n = 14, as we have to find the sum of the first 14 terms. And from the given AP, we can see that the first term is 12. So, we get, a = 12. We know that the common difference of an AP is the difference between the consecutive pairs of the AP. Hence, common difference is given by,
d = (15-12) = (18-15) = (21-18) = 3
So, we get the common difference, d = 3.
Now, by substituting these values in the formula, ${{S}_{n}}=\dfrac{n}{2}\left( 2a+\left( n-1 \right)d \right)$, we get,
$\begin{align}
  & {{S}_{14}}=\dfrac{14}{2}\left( 2\left( 12 \right)+\left( 14-1 \right)3 \right) \\
 & =7\left( 24+\left( 13 \right)3 \right) \\
 & =7\left( 24+39 \right) \\
 & =7\times 63 \\
 & =441 \\
\end{align}$
Hence, we get the sum of the first 14 terms of the AP 12, 15, 18, 21 …… as 441.
Hence, option (C) is the correct answer.

Note: In this question, the given options are a little confusing, so the students must choose the correct option very carefully. Also remember that while using the formula of the sum of n terms of an AP, we should use 2a and not just ‘a’ as it is a common mistake, as the students tend to write ‘a’ instead of 2a. So, students must know that the correct formula to find the sum on n terms of an AP is, ${{S}_{n}}=\dfrac{n}{2}\left( 2a+\left( n-1 \right)d \right)$.