Question

Which term of A.P. \[92,88,84,80,....\] is 0?
A. 24
B. 32
C. 33
D. 12

Answer 1

Hint: Find the common difference of the given arithmetic series.
In the given series first we will find the common difference of the series and since the first term of the series is given so we will find the nth term of the series assuming that 0 is the nth term of the series by applying the formula \[{T_n} = a + \left( {n - 1} \right)d\], where \[a\] is the first term of the series, \[d\] is the common difference and \[n\] is the term which we are finding.
When we will substitute the values in the formula we will get the term of \[0\].

Complete step by step answer:
Given the series \[92,88,84,80,....\]
Now since the given series is Arithmetic progressive series so we will find its common difference which is the difference between the each term with its progressive term denoted by the term \[d\], so we can write
\[
  d = 88 - 92 = - 4 \\
  d = 84 - 88 = - 4 \\
  d = 80 - 84 = - 4 \\
 \]
Hence the common difference of the series is \[d = - 4\], difference is negative since the series is an arithmetically decreasing series.
From the series we can see the first term of the series \[a = 92\]
Now let us assume that \[0\] whose term is to be find is the nth term of the series \[{T_n} = 0\] and since we know the formula \[{T_n} = a + \left( {n - 1} \right)d\] to find the nth term of the series, hence by substituting we can write
\[
  {T_n} = a + \left( {n - 1} \right)d \\
  0 = 92 + \left( {n - 1} \right)\left( { - 4} \right) \\
 \]
By further solving this, we get
\[
  92 - 4\left( {n - 1} \right) = 0 \\
   \Rightarrow 92 - 4n + 4 = 0 \\
   \Rightarrow 96 - 4n = 0 \\
   \Rightarrow 4n = 96 \\
   \Rightarrow n = \dfrac{{96}}{4} \\
   \Rightarrow n = 24 \\
 \]
Hence we can say \[0\] is the 24th term of the series.

Hence option A is correct.

Note: Difference between two consecutive terms of an arithmetic progression is known as common difference and it tells the nature of the progression. If the common difference of a series is \[d > 0\], then the series is arithmetically increasing and if the \[d < 0\] then the series is arithmetically decreasing. In this question if the common difference would have not been negative then the series would not have any term 0.