Question

How many terms of the A.P. $3,6,9,12,15...$must be taken to make the sum $108$?
A. $6$
B. $7$
C. $8$
D. $36$

Answer 1

The numbers of a series are said to be in A.P if and only if the common difference (that is, the difference between the consecutive terms) remains constant throughout the series. Next, use the direct formula for the sum of $n$ terms of an AP and then find out the value of $n$ for the given sum.

Complete step by step answer:
The given AP is $3,6,9,12,15...$. The first term (a) of the A.P is $3$. The common difference (d) of the A.P is $3$
$\Rightarrow d = \left( {6 - 3} \right) = \left( {9 - 6} \right) = 3$
We have to find out the number of terms in the A.P whose sum equals to be $108$.
As we already know that the sum (${S_n}$) of an A.P is given by:
${S_n} = \dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)$
Here substituting the values of $a = 3$ and $d = 3$, we get,
$\Rightarrow 108 = \dfrac{n}{2}\left( {2 \times 3 + \left( {n - 1} \right)3} \right)$
Now by simplifying the above equation we get,
$\Rightarrow 108 = \dfrac{n}{2}\left( {6 + \left( {n - 1} \right)3} \right) \\\ \Rightarrow 108 = \dfrac{n}{2}\left( {6 + 3n - 3} \right) \\\ \Rightarrow 108 = \dfrac{n}{2}\left( {3n + 3} \right) \\\ \Rightarrow 216 = 3{n^2} + 3n \\\ \Rightarrow 3{n^2} + 3n - 216 = 0 \\\$
Now, we have a quadratic equation so we apply the quadratic formula to find out the value of $n$.
$n = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$...............(where $a = 3,b = 3,c = - 216$)
$\Rightarrow n = \dfrac{{ - 3 \pm \sqrt {{3^2} - 4\left( 3 \right)\left( { - 216} \right)} }}{{2\left( 3 \right)}} \\\ \Rightarrow n = \dfrac{{ - 3 \pm \sqrt {9 + 2592} }}{6} \\\ \Rightarrow n = \dfrac{{ - 3 \pm \sqrt {2601} }}{6} \\\ \Rightarrow n = \dfrac{{ - 3 \pm 51}}{6} \\\ \Rightarrow n = \dfrac{{ - 3 + 51}}{6},\dfrac{{ - 3 - 51}}{6} \\\ \Rightarrow n = \dfrac{{48}}{6},\dfrac{{ - 54}}{6} \\\$
As we know that a negative value of $n$ is not possible. So,
$\therefore n = \dfrac{{48}}{6} = 8$
So, we are required to take $8$ terms in an A.P to give a sum of $108$.

Hence, $n = 8$ is the required answer and option C is correct.

Note: Whenever faced with such types of questions, it is advised to have good knowledge of general formulas related to the series which is asked in the question. Here in order to solve this question, we must have knowledge of arithmetic progression. This will not only help save a lot of time but also to get the right answer.