Question

Which term of the progression \[20,19\dfrac{1}{4},18\dfrac{1}{2},17\dfrac{3}{4},.....\] is the first negative term?

Answer 1

Hint: First we need to check if the given series is in Arithmetic progression or in Geometric progression or in Harmonic progression. We can check by taking the common difference or common ratio of the adjacent numbers or terms in the series. By knowing the formula for the \[{n^{th}}\] term of the series we can solve this.

Complete step-by-step answer:
Given,
 \[20,19\dfrac{1}{4},18\dfrac{1}{2},17\dfrac{3}{4},.....\] Here first term is \[a = 20\] .
Let’s check first for common difference,
That is \[d = 19\dfrac{1}{4} - 20\]
Converting mixed fraction into improper fraction,
 \[ \Rightarrow \dfrac{{76 + 1}}{4} - 20\]
 \[ \Rightarrow \dfrac{{77}}{4} - 20\]
Taking L.C.M. we have,
 \[ \Rightarrow \dfrac{{77 - 80}}{4}\]
 \[ \Rightarrow - \dfrac{3}{4}\] .
Let’s check for another one to confirm is it same or not,
 \[d = 18\dfrac{1}{2} - 19\dfrac{1}{4}\]
Converting mixed fraction into improper fraction,
 \[ \Rightarrow \dfrac{{36 + 1}}{2} - \dfrac{{76 + 1}}{4}\]
 \[ \Rightarrow \dfrac{{37}}{2} - \dfrac{{77}}{4}\]
Taking L.C.M. we have,
 \[ \Rightarrow \dfrac{{74 - 77}}{4}\]
 \[ \Rightarrow - \dfrac{3}{4}\]
Hence we can see that the common difference is \[d = - \dfrac{3}{4}\]
Hence the given series is in Arithmetic progression, because the common difference between two adjacent numbers is the same.
We know the formula for \[{n^{th}}\] term in A.P is \[{T_n} = a + (n - 1)d\] .
Let consider that \[{n^{th}}\] in A.P is the first negative number, then
 \[{a_n} < 0\]
 \[ \Rightarrow a + (n - 1)d < 0\]
We have, \[a = 20\] and \[d = - \dfrac{3}{4}\] . Substituting we have,
 \[ \Rightarrow 20 + (n - 1) \times \left( { - \dfrac{3}{4}} \right) < 0\]
 \[ \Rightarrow 20 - \dfrac{{3n}}{4} + \dfrac{3}{4} < 0\]
 \[ \Rightarrow 20 + \dfrac{3}{4} - \dfrac{{3n}}{4} < 0\]
Taking L.C.M. we have,
 \[ \Rightarrow \dfrac{{80 + 3 - 3n}}{4} < 0\]
 \[ \Rightarrow \dfrac{{83 - 3n}}{4} < 0\]
 \[ \Rightarrow 83 - 3n < 0\]
 \[ \Rightarrow 83 < 3n\]
We can write it as,
 \[ \Rightarrow 3n > 83\]
Divided by 3 on both sides we have,
 \[ \Rightarrow n > \dfrac{{83}}{3}\]
 \[ \Rightarrow n > 27.66666\]
Rounding off we have
 \[ \Rightarrow n \geqslant 28\]
That is, the 28th term is the first negative term of the given A.P.
So, the correct answer is “28th term”.

Note: In above if the given series have different common differences then check for the common ratio. If we have the same common ratio then it is in geometric progression. The formula for \[{n^{th}}\] term in G.P. is \[{T_n} = a.{r^{n - 1}}\] . Where ‘r’ is the common ratio. Follow the same procedure as above. Also note that in the above series they are given in mixed fraction, so we need to convert into improper fraction while finding the common ratio or common difference.