Question

which term of the ap 19,181/5,172/5,....... is first negative term

Answer 1

First we must find the common difference of the A.P-  

Common difference (d) = 2nd term- 1st term  

= 18 1/5 - 19  
= 91/5 - 95/5  
= - 4/5  

Hence, the common difference is -4/5 or - 0.8  

Now to estimate which term of this A.P will be negative let us assume the nth term of the AP is 0.  

Formula for nth term of an A.P = a+ (n-1) d  

Then,  

0 = 19 + (n-1) * - 0.8  
0 = 19 - 0.8n + 0.8  
0.8n = 19.8  
n = 19.8/ 0.8  
n = 24.75  

If the 24.75 th term is 0. Then surely the 25th term of the AP will be negative.  

Let us make sure, just in case:  

25th term of an AP= a + (n-1)d  
= 19 + (25-1) * - 0.8  
= 19 + (24 * - 0.8)  
= 19 + (- 19.2)  
= 19 - 19.2  
= - 0.2 

In case you still don't believe me, let me verify that the 24th term is positive-  

24th term= a+ (n-1)d  
= 19 + (24-1) * - 0.8  
= 19 + (23 * - 0.8) 
= 19 - 18.4  
= 0.6  
 
Thus, the 25th term of the of the progression 19,18 1/5,17 2/5... is the first negative.  

Hope it helps :)