Question

Which term of the A.P: $ 53,48,43... $ is the first negative term?

Answer 1

Hint: In this question we have to find the term which will be the negative term. So first we will calculate the common difference with the given A.P by subtracting any two consecutive terms. We will assume the first negative term is the $ {n^{th}} $ term. After this we will use the formula $ {a_n} = a + (n - 1)d $ to get the $ {n^{th}} $ term of the A.P.

Complete step-by-step answer:
As per the question we have the AP: $ 53,48,43... $ . The general form of the AP is $ a,{a_1},{a_2}...{a_n} $ .
Here the first term i.e. $ a = 53 $ . Now the common difference i.e.
$ d = 48 - 53 = - 5 $ .
Let the $ n $ th term of the given AP be the first negative term. It means that it is less than zero. We can write it as $ {a_n} < 0 $ .
Now by putting the formula we can write it as $ a + (n - 1)d < 0 $ . We will put the values of the terms in this expression and we can write it as
$ 53 + (n - 1) (- 5) < 0 $ .
On further solving we can write it as
$ (n - 1) \times - 5 < - 53 \Rightarrow n - 1 < \dfrac{{ - 53}}{{ - 5}} $ .
By isolating the term $ n $ , we have
$ n < \dfrac{{53}}{5} + 1 \Rightarrow n < \dfrac{{53 + 5}}{5} $ .
It gives us the value $ n < \dfrac{{58}}{5} $ . Here we have to keep in mind that the value of $ n $ should be such that it has to be more than $ \dfrac{{58}}{5} = 11.6 $ . The number greater than this starts from $ 12 $ .
 So the least possible value in the place of $ n $ which satisfies the given equation is $ 12 $ . Therefore $ n = 12 $ .
Hence the required term is $ 12th $ which is the first negative term.
So, the correct answer is “ $ 12th $ ”.

Note: We should note that in inequality when a positive term moves to the other side, it turns into the negative term. We should know that arithmetic progression (AP) is a sequence of numbers in which the difference between any two consecutive numbers is a constant. We must remember that the value of $ n $ is always a positive integer.