Question

Which term of the A.P. $1,4,7.........$ is 88?
(a) 26
(b) 27
(c) 30
(d) 35

Answer 1

Hint: For solving this question first we will understand what we mean by arithmetic progression and then we will see the formula for the ${{n}^{th}}$ term of an arithmetic progression is $a+\left( n-1 \right)d$ where, $a$ is the first term and $d$ is the common difference. After that, we will equate $a+\left( n-1 \right)d$ to 88 and solve for the value of $n$.

Complete step-by-step answer:
Given:
We have an arithmetic progression 1, 4, 7, ……… and we have to find which term of the given arithmetic progression is 88.
Now, first, we will understand when a sequence is called an A.P. and what are important conditions for a sequence to be in arithmetic progression.
Arithmetic Progression:
In a sequence when the difference between any two consecutive terms is equal throughout the series then, such sequence will be called to be in arithmetic progression and the difference between consecutive terms is called as the common difference of the arithmetic progression. If ${{a}_{1}}$ is the first term of an A.P. and common difference of the A.P. is $d$ then, ${{n}^{th}}$ the term of the A.P. can be written as ${{a}_{n}}={{a}_{1}}+d\left( n-1 \right)$.
Now, we come back to our question in which we have A.P. $1,4,7........$. And it is evident that the difference between the consecutive terms is equal to $4-1=7-4=3$ and 3 is the first term. So, here the value of ${{a}_{1}}=1$ and $d=3$ .
Now, let ${{a}_{n}}$ is the ${{n}^{th}}$ term of the A.P. and ${{a}_{1}}=1$ , $d=3$ . Then,
$\begin{aligned} & {{a}_{n}}={{a}_{1}}+d\left( n-1 \right) \\\ & \Rightarrow {{a}_{n}}=1+3\left( n-1 \right) \\\ & \Rightarrow {{a}_{n}}=3n-2 \\\ \end{aligned}$
Now, equate the expression of ${{n}^{th}}$ term of the A.P. ${{a}_{n}}=3n-2$ to 88. Then,
$\begin{aligned} & {{a}_{n}}=3n-2=88 \\\ & \Rightarrow 3n=90 \\\ & \Rightarrow n=30 \\\ \end{aligned}$
Now, from the above result, we conclude that 88 will be the ${{30}^{th}}$ term of the given arithmetic progression.
Hence, (c) is the correct option.
Note: Here, the student should know the concept of A.P. and how to express the general expression of ${{n}^{th}}$ the term of an A.P. and important point should be remembered so, that question can be answered quickly and correctly without any confusion.