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Question: Which term of the AP \( 40,35,30,.... \) is the first negative term? \( A)9^{th} \) \( B)10^...

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

Explanation

Solution

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),...a,(a + d),(a + 2d),(a + 3d),... where aa is the first term and dd is the common difference. But in the

Complete step by step answer:
Formula to consider for solving these questions an=a+(n−1)d{a_n} = a + (n - 1)d
Where dd is the common difference, aa 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,....40,35,30,....
Since the first term in the problem is 4040 which is the aa as well as t1=40{t_1} = 40 is the starting value too,
The second value is 3535 so then second value subtracts the first values gives as the common difference
Thus t2−t1=35−40{t_2} - {t_1} = 35 - 40 =−5= - 5 similarly for the third value given is 3030 and then 35−30=−535 - 30 = - 5 which is the third term subtracts the second term of t3−t2=−5{t_3} - {t_2} = - 5
Hence the common difference for all two compared values is −5- 5
Now we need to find the first negative term in 40,35,30,....40,35,30,....
Since the first term is 4040 and the second term is 3535 , the third term is 3030 similarly as approaching the same way further we get for
Fourth term, fifth term, sixth term, seventh term, eighth term, ninth term, as 25,20,15,10,5,025,20,15,10,5,0
Respectively (the common difference is -5 so by AP formula we get all these)
Hence at the 10th{10^{th}} term we get 0−5=−50 - 5 = - 5 (ninth term is zero and minus that we get)
Which is the 1st{1^{st}}negative term in the given AP 40,35,30,....40,35,30,.... and it is occupied in the 10th{10^{th}} term is negative.

So, the correct answer is “Option B”.

Note: Since for option A the 9th{9^{th}} term is zero, for option C the 12th12^{th} term is −15- 15 and the option D the 14th{14^{th}} term is −25- 25 but in question they were asking the first negative and hence it is in the 10th{10^{th}} term
So other options except B are eliminated.