Question

Which term of the A.P. $101,97,.........$ is its first negative term?
A.26
B.25
C.28
D.27

Answer 1

Hint: At first we will assume that the ${n^{th}}$ term is the first negative term of A.P., then we will solve for the inequality on solving for our assumption.
The minimum value of ‘n’ satisfying the inequality will be the required value of ‘n’ and hence we will get the answer.

Complete step-by-step answer:
Given data: An A.P. i.e. $101,97,.........$
We know if we have an A.P. ${a_1},{a_2},{a_3},..............$ then the common difference ‘d’ is given by
$d = {a_2} - {a_1} = {a_3} - {a_2}.....$ as the definition of common difference says that it is the difference of a term from its preceeding term in an A.P. as it is constant throughout an A.P. and the general term is given by
${a_n} = {a_1} + (n - 1)d$
Now according to the given A.P., we have ${a_1} = 101$ , and ${a_2} = 97$
Therefore, $d = {a_2} - {a_1}$
$ = 97 - 101$
$\therefore d = - 4$
Now let the ${n^{th}}$ term is the first negative term i.e. ${a_n} < 0$
Substituting the value of ${a_n}$
$ \Rightarrow {a_1} + (n - 1)d < 0$
Substituting the value of ${a_1}$, and $d$
$ \Rightarrow 101 - 4(n - 1) < 0$
On adding $4(n - 1)$both the sides we get,
$ \Rightarrow 101 < 4(n - 1)$
On simplifying the brackets, we get
$ \Rightarrow 101 < 4n - 4$
Adding 4 on both sides and then dividing by 4, we get
$ \Rightarrow \dfrac{{105}}{4} < n$
On simplification we get,
$ \Rightarrow 26.25 < n$
Since $26.25 < n$ the term will be negative and the minimum value of ‘n’ will be the first negative term
Therefore the required value of n is 27
Option(D) is correct.

Note: Most of the students will try to create another inequality by solving it for ${(n - 1)^{th}}$ term but solving it we will get several values ‘n’, since solving the inequality we have calculated can be satisfactory for the answer we do not need another inequality for the answer.