Question

Which term of progression 20,77/4,37/2,71/4is the first negative term

Answer 1

d=-3/4
a=20
therefore
nth term
a+(n-1)d<0
20+(n-1)-3/4<0
20+3/4-3n/4<0
83/4-3n/4<0
-3n<-83
n~=27.6
therefore first term negative will be 28

Answer 2

20 , 77/4 ,37/2 ,  71/4

This could be written as 80/4 , 77/4 , 74/4 , 71/4

This an AP with the first term 80/4 and common difference -3/4

forget about the denominator for a while as positive and negative sign depends on the numerator. In every step 3 is being subtracted from the numerator.

80/3 = 26.67

So if we subract 27d from the first term it will give us a negative no

So the 28th term will be the first negative term
a28= 80+ 27d

a28 = 80/4 +27( -3/4)= -1/4