Question

Find the number of terms in the sequence 4, 12, 20…..108.
(a) 12
(b) 19
(c) 13
(d) 14

Answer 1

If you carefully look at the first three terms of an A.P., you will find out that these three terms have a common difference of 8. The common difference is calculated by subtracting any number from its succeeding term. Now, we know the general term for the A.P. which is equal to: ${{T}_{n}}=a+\left( n-1 \right)d$. Here, $''a''$ is the first term, “d” is a common difference. And n is the order of the term (like third, fourth). After that, substitute ${{T}_{n}}$ as 108 in the general term and find the value of n. This value of n will give you the number of terms.

Complete step by step answer:
The sequence given in the above problem is as follows:
4, 12, 20…..108
The first term $\left( a \right)$ of the above sequence is 4 and the common difference of the above sequence is 8.
The common difference we are going to calculate by subtracting any number (say 12) from its succeeding term (say 20) and we get,
$20-12=8$
Now, we know the general term of an A.P. as follows:
${{T}_{n}}=a+\left( n-1 \right)d$
Substituting the value of $a=4$ and $d=8$ in the above equation we get,
${{T}_{n}}=4+\left( n-1 \right)8$
Now, to find the number of terms in the given sequence we are going to substitute ${{T}_{n}}=108$ in the above equation we get,
$108=4+\left( n-1 \right)8$
Multiplying 8 with the bracket we get,
$\begin{aligned} & \Rightarrow 108=4+8n-8 \\\ & \Rightarrow 108=8n-4 \\\ \end{aligned}$
Adding 4 on both the sides of the above equation we get,
$\begin{aligned} & 108+4=8n \\\ & \Rightarrow 112=8n \\\ \end{aligned}$
Dividing 8 on both the sides of the above equation we get,
$\begin{aligned} & \dfrac{112}{8}=n \\\ & \Rightarrow n=14 \\\ \end{aligned}$
From the above, we have found the total number of terms in the given sequence as 14.

So, the correct answer is “Option d”.

Note: To solve the above problem, you must be familiar with the concepts of the general term of an A.P. and how to calculate the common difference of an A.P. Also, while finding the common difference of an A.P., make sure you have subtracted any term from its succeeding term not subtracting any term from its preceding term.