Question

Which term of the AP $ 40,35,30,.... $ is the first negative term?
 $ A)9^{th} $
 $ B)10^{th} $
 $ C)12^{th} $
 $ D)14^{th} $

Answer 1

Hint: First we have to define what the terms we need to solve the problem are.
The given question they were asking to find the first negative term, solution as follows, since we need to know about Arithmetic progression.
Formula used:
An arithmetic progression can be given by $ a,(a + d),(a + 2d),(a + 3d),... $ where $ a $ is the first term and $ d $ is the common difference. But in the

Complete step by step answer:
Formula to consider for solving these questions $ {a_n} = a + (n - 1)d $
Where $ d $ is the common difference, $ a $ is the first term, since we know that difference between consecutive terms is constant in any A.P
Let us find the first negative in given terms of AP $ 40,35,30,.... $
Since the first term in the problem is $ 40 $ which is the $ a $ as well as $ {t_1} = 40 $ is the starting value too,
The second value is $ 35 $ so then second value subtracts the first values gives as the common difference
Thus $ {t_2} - {t_1} = 35 - 40 $ $ = - 5 $ similarly for the third value given is $ 30 $ and then $ 35 - 30 = - 5 $ which is the third term subtracts the second term of $ {t_3} - {t_2} = - 5 $
Hence the common difference for all two compared values is $ - 5 $
Now we need to find the first negative term in $ 40,35,30,.... $
Since the first term is $ 40 $ and the second term is $ 35 $ , the third term is $ 30 $ similarly as approaching the same way further we get for
Fourth term, fifth term, sixth term, seventh term, eighth term, ninth term, as $ 25,20,15,10,5,0 $
Respectively (the common difference is -5 so by AP formula we get all these)
Hence at the $ {10^{th}} $ term we get $ 0 - 5 = - 5 $ (ninth term is zero and minus that we get)
Which is the \[{1^{st}}\]negative term in the given AP $ 40,35,30,.... $ and it is occupied in the $ {10^{th}} $ term is negative.

So, the correct answer is “Option B”.

Note: Since for option A the \[{9^{th}}\] term is zero, for option C the $ 12^{th} $ term is $ - 15 $ and the option D the \[{14^{th}}\] term is $ - 25 $ but in question they were asking the first negative and hence it is in the $ {10^{th}} $ term
So other options except B are eliminated.