Question

Which term of A.P. $20,19\dfrac{1}{4},18\dfrac{1}{2},...........$ is the first negative term?
A. $18th$
B. $15th$
C. $28th$
D. $27th$

Answer 1

Hint: In order to find the first negative term, first we will find the number of terms by applying the nth term formula as ${a_n} = a + \left( {n - 1} \right)d$, where
 $
  {a_n} = {\text{ nth term }} \\
  n = {\text{number of terms}} \\
  d = {\text{ common difference}} \\
 $

Complete Step-by-Step solution:
Given series is $20,19\dfrac{1}{4},18\dfrac{1}{2},...........$
Here first term $a = 20$
Now, we will evaluate the common difference
$
  {a_2} - {a_1} = {a_3} - {a_2} \\
   \Rightarrow {a_2} - {a_1} = 19\dfrac{1}{4} - 20 = \dfrac{{ - 3}}{4} \\
   \Rightarrow {a_3} - {a_2} = 18\dfrac{1}{2} - 19\dfrac{1}{4} = \dfrac{{ - 3}}{4} \\
  \therefore {\text{ common difference, d}} = \dfrac{{ - 3}}{4} \\
 $
Let ${a_n}$ be the first negative term of given series
$\therefore {a_n} < 0$
We know that if our series is in A.P. then nth term can be expressed as ${a_n} = a + \left( {n - 1} \right)d,$
By substituting the value of ${a_n},$ we have
${a_n} = a + \left( {n - 1} \right)d < 0$
Put the value of a and d
\[
   \Rightarrow 20 + \left( {n - 1} \right)\left( {\dfrac{{ - 3}}{4}} \right) < 0 \\
   \Rightarrow \dfrac{{80 - 3n + 3}}{4} < 0 \\
   \Rightarrow 80 - 3n + 3 < 0 \\
   \Rightarrow - 3n < - 83 \\
   \Rightarrow 3n > 83 \\
   \Rightarrow n > \dfrac{{83}}{3} = 27.66 \\
 \]
Since, n is a natural number n =28.
Now, we have a, d and n, we will proceed further by applying nth term formula to get the first negative term.
$\therefore {a_n} = a + \left( {n - 1} \right)d$
Substitute the value of a, n and d
$
   \Rightarrow {a_n} = 20 + \left( {28 - 1} \right)\left( {\dfrac{{ - 3}}{4}} \right) \\
   \Rightarrow {a_n} = 20 + \left( {27} \right)\left( {\dfrac{{ - 3}}{4}} \right) \\
   \Rightarrow {a_n} = \dfrac{{80 - 81}}{4} \\
   \Rightarrow {a_n} = \dfrac{{ - 1}}{4} \\
 $
Hence, $\dfrac{{ - 1}}{4}$ is the first negative term of a given series. And the correct option is “C”.

Note: In order to solve these types of problems, you need to remember the formula of the nth term of the A.P. series. Also remember the formula of sum of n terms of A.P. series. The most problems are based on finding the nth term with some conditions given in the question such as in the current problem we have to find the value of the first negative term in the given series.